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PreCal B

Student Name: __________________

Student ID: _____________________

School Name: ___________________

Summer School Distance

Learning Packet

Teacher Name: __________________

High School Math

Brownsville Independent School District

1900 Price Road, Brownsville, TX 78521, (956) 548-8000 www.bisd.us

May 2020

Esteemed Parents and Family Members,

We hope this letter finds you safe and healthy amid this uneasy time of COVID-19. As always, our priority is

the safety and welfare of our students. Our 2020 summer program will continue by utilizing virtual learning

platforms and will begin on June 1 and end on June 18, 2020. The purpose of the summer program is to

provide students the opportunity to gain credit for the course your student has failed.

You have received this summer 2020 instructional packet for your (9th - 12th grade) student. This instructional

packet includes materials for the core area(s) your student has failed.

We ask that you contact your student’s school to:

• give you the failing subject area(s)

• give you your student’s summer teachers’ name and contact information / email address

• update any contact information including any changes and additional contact numbers, and

email addresses, etc.

• receive login information for the digital platform

The platform utilized this summer will be:

• 9th -12th Google Classroom

(Download Google Classroom app or access through the Clever Portal)

Our sincere hope is that your child will participate and take advantage of this opportunity for promotion that

will greatly support your child’s area(s) of educational need.

Please encourage your student to read, watch educational programs, and practice their writing and speaking

skills. This is also a great time to share family stories and traditions, play board games and enjoy family time.

As always, it is an honor to continue to serve you and we value your family's commitment in entrusting us with

your child's education.

BISD does not discriminate on the basis of race, color, national origin, gender, religion, age, or disability or genetic information in employment

or provision of services, programs, or activities.

Brownsville Independent School District

1900 Price Road, Brownsville, TX 78521, (956) 548-8000 www.bisd.us

Mayo de 2020

Estimados Padres y Miembros de Familia,

Esperamos que esta carta le encuentre a buen resguardo y en buena salud durante estos días difíciles del

COVID-19. Como siempre, nuestra prioridad es la seguridad y el bienestar de nuestros estudiantes. Nuestro

programa de verano 2020 continuará utilizando plataformas de aprendizaje virtuales y comenzará el 1 de junio

y terminará el 18 de junio de 2020. El propósito del programa de verano es proporcionar a los estudiantes que

no fueron promovidos al siguiente grado, una oportunidad para obtener la promoción.

Con el fin de trabajar en la promoción de su hijo/a al siguiente grado, usted ha recibido un paquete de

instrucción para el verano del 2020 para su hijo/a de preparatoria. Dicho paquete incluye materiales para la(s)

asignatura(s) que su hijo/a reprobó.

Le pedimos que se ponga en contacto con la escuela de su hijo/a para:

• darle el área(s) de materia(s) que está reprobando.

• darle el nombre del maestro/a de verano de su hijo/a y su correo electrónico

• actualizar cualquier información de contacto, incluyendo cualquier cambio y números

de contacto adicionales, y correo electrónico, etc.

• recibir la información para conectarse a las plataformas digitales

La siguiente plataforma virtual se utilizará este verano para la preparatoria:

• Google Classroom

(Descargar aplicación de Google Classroom o usar el portal de Clever)

Esperamos sinceramente que su hijo/a participe y aproveche esta oportunidad de promoción que apoyará en

gran medida las áreas de su necesidad educativa.

Anime a sus hijos/as a leer, ver programas educativos y practicar sus habilidades para escribir y hablar. Este es

también un gran momento para compartir historias y tradiciones familiares, jugar juegos de mesa y disfrutar

del tiempo en familia.

Como siempre, es un honor continuar sirviéndole y valoramos nuestro compromiso con su familia al

confiarnos la educación de su hijo/a.

BISD no discrimina de acuerdo de raza, color, origen nacional, género, religión, edad, información genética, o incapacidad en el empleo o en

la provisión de servicios, programas o actividades.

Pre-Cal-B Summer 2020

Date Resources

Week 1 (June 1-June 5) Ch 4 L 1 Right Triangle Trigonometry Chapter 4 Lesson 1

Ch 4 L2 Degrees and Radians Chapter 4 Lesson 2

Ch 4 L3 Trigonometric Functions on the Unit Circle Chapter 4 Lesson 3

Ch 4 L4 Graphing Sine and Cosine Functions Chapter 4 Lesson 4

Ch 4 L5 Graphing Other Trigonometric Functions Chapter 4 Lesson 5

Ch 4 L6

Inverse Trigonometric Functions Chapter 4 Lesson 6

Week 3 (June 15-18) Ch 4 L7 The Law of Sines and the Law of Cosines Chapter 5 Lesson 1

Ch 5 L1 Chapter 5 Lesson 2

Finalize all Assignments

PreCal-B

Student Name: __________________

Student ID: _____________________

School Name: ___________________

Summer School 2020

Week 1

Teacher Name: __________________

High School Math

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(continued on the next page)

This is an alphabetical list of key vocabulary terms you will learn in Chapter 4. As you study this chapter, complete each term’s definition or description. Remember to add the page number where you found the term. Add these pages to your Precalculus Study Notebook to review vocabulary at the end of the chapter.

Vocabulary TermFound

on PageDefi nition/Description/Example

amplitude

angle of depression

angle of elevation

cosecant

cosine

coterminal angle

frequency

Law of Cosines

Law of Sines

period

Student-Built Glossary4

Chapter 4 1 Glencoe Precalculus

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Vocabulary TermFound

on PageDefi nition/Description/Example

phase shift

radian

reference angle

secant

sine

standard position

tangent

unit circle

vertical shift

Student-Built Glossary4



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4

Before you begin Chapter 4

• Read each statement.

• Decide whether you Agree (A) or Disagree (D) with the statement.

• Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure).

After you complete Chapter 4

• Reread each statement and complete the last column by entering an A or a D.

• Did any of your opinions about the statements change from the first column?

• For those statements that you mark with a D, use a piece of paper to write an example of why you disagree.

STEP 1 A, D, or NS

StatementSTEP 2 A or D

1. If a trigonometric ratio is given in a quadrant, the remaining five trigonometric ratios can be found.

2. The Law of Sines can only be used with right triangles. 3. The area of a sector of a circle can be found if the radius and

a central angle are known. 4. One full rotation on the unit circle is π radians. 5. The amplitude of a sinusodial function is the distance from

the highest point to the lowest point. 6. The rate at which an object moves along a circular path is

called its linear speed. 7. The period of a sinusoidal function refers to the number of

cycles the function completes in a one unit interval. 8. Trigonometric functions have inverses. 9. The sine of an acute angle is the ratio of the measure of the

side opposite the angle in a right triangle to the measure of the side adjacent to the angle.

10. Heron’s formula can be used to find the perimeter of any triangle.

11. The graph of the sine function has vertical asymptotes at odd

multiples of π − 2 .

12. A damped trigonometric function can be used to model the vibrations of a guitar string.

Anticipation GuideTrigonometric Identities and Equations

Step 2

Step 1



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Antes de que comiences el Capítulo 4

• Lee cada enunciado.

• Decide si estás de acuerdo (A) o en desacuerdo (D) con el enunciado.

• Escribe A o D en la primera columna O si no estás seguro(a), escribe NS (no estoy seguro(a)).

Después de que termines el Capítulo 4

• Relee cada enunciado y escribe A o D en la última columna.

• Compara la última columna con la primera. ¿Cambiaste de opinión sobre alguno de los enunciados?

• En los casos en que hayas estado en desacuerdo con el enunciado, escribe en una hoja aparte un ejemplo de por qué no estás de acuerdo.

PASO 1 A, D o NS

EnunciadoPASO 2 A o D

1. Dada una razón trigonométrica, se pueden calcular las otras cinco razones trigonométricas.

2. La ley de los senos sólo se puede aplicar a triángulos rectángulos.

3. El área de un sector de un círculo se puede calcular si se conocen el radio y un ángulo central del círculo.

4. Una rotación completa alrededor del círculo unitario equivale a π radianes.

5. La amplitud de una función sinusoidal es igual a la distancia entre su punto más alto y su punto más bajo.

6. La tasa a la cual un cuerpo recorre un trayecto circular se conoce como su velocidad lineal.

7. El período de una función sinusoidal se refiere al número de ciclos que la función completa por cada intervalo unidad.

8. Las funciones trigonométricas tienen inverso. 9. El seno de un ángulo es la razón de la medida del lado

opuesto al ángulo de un triángulo rectángulo a la medida del ángulo adyacente a dicho ángulo.

10. La fórmula de Herón sirve para calcular el perímetro de cualquier triángulo.

11. La gráfica de una función seno tiene asíntotas verticales en múltiplos impares de π − 2 .

12. Las vibraciones de una cuerda de guitarra se pueden representar por una función trigonométrica amortiguada.

Ejercicios preparatoriosIdentidades y ecuaciones trigonométricas

4

Paso 2

Paso 1

Capítulo 4 4 Precálculo de Glencoe


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Values of Trigonometric Ratios The side lengths of a right triangle and a reference angle θ can be used to form six trigonometric ratios that define the trigonometric functions known as sine, cosine, and tangent. The cosecant, secant, and cotangent ratios are reciprocals of the sine, cosine, and tangent ratios, respectively. Therefore, they are known as reciprocal functions.

Find the exact values of the six trigonometric

functions of θ.

Use the Pythagorean Theorem to determine the length of the hypotenuse. 152 + 32 = c2 a = 15, b = 3

234 = c2 Simplify.

c = √ �� 234 or 3 √ �� 26 Take the positive square root.

sin θ = opp

− hyp

= 3 −

3 √ �� 26 or

√ �� 26 −

26 cos θ =

adj −

hyp = 15 −

3 √ �� 26 or 5 √ �� 26

− 26

tan θ = opp

− adj

= 3 − 15

or 1 − 5

csc θ = hyp

− opp = 3 √ �� 26 −

3 or √ �� 26 sec θ =

hyp −

adj = 3 √ �� 26

− 15

or √ �� 26

− 5 cot θ =

adj − opp = 15 −

3 or 5

ExercisesFind the exact values of the six trigonometric functions of θ.

1. 10

5θ

2. 19

7

θ

Use the given trigonometric function value of the acute angle θ to find the exact values of the five remaining trigonometric function values of θ.

3. sin θ = 3 − 7

Example

Study Guide and InterventionRight Triangle Trigonometry

4-1

Let θ be an acute angle in a right triangle and the abbreviations opp, adj, and hyp refer

to the lengths of the side opposite θ, the side adjacent to θ, and the hypotenuse, respectively.

Then the six trigonometric functions of θ are defi ned as follows.

sine (θ) = sin θ = opp

− hyp

cosine (θ) = cos θ = adj

− hyp

tangent (θ) = tan θ = opp

− adj

cosecant (θ) = csc θ = hyp

− opp

secant (θ) = sec θ = hyp

− adj

cotangent (θ) = cot θ = adj

− opp

θ3

15

opp

adj

hyp

θ

4. sec θ = 8 − 5

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Solving Right Triangles To solve a right triangle means to find the measures of all of the angles and sides of the triangle. When the trigonometric value of an acute angle is known, the inverse of the trigonometric function can be used to find the measure of the angle.

Trigonometric Function Inverse Trigonometric Function

y = sin xy = cos xy = tan x

x = sin-1 y or x = arcsin yx = cos-1 y or x = arccos yx = tan-1 y or x = arctan y

Solve � ABC. Round side measures to the nearest tenth and angle measures to the nearest degree.

Because two lengths are given, you can use the Pythagorean Theorem to find that a is equal to √ �� 825 or about 28.7.Find the measure of ∠A using the cosine function.

cos θ = adj

− hyp

Cosine function

cos A = 20 − 35

Substitute b = 20 and c = 35.

A = cos -1

20 − 35

Defi nition of inverse cosine

A = 55.15009542 Use a calculator.

Because A is now known, you can find B by subtracting A from 90°. 55.15 + B = 90 Angles A and B are complementary.

B = 34.85° Subtract.

Therefore, a ≈ 28.7, A ≈ 55°, and B ≈ 35°.

Exercises

Find the value of x. Round to the nearest tenth if necessary.

1.

20x

60°

2. 8

x

25°

Solve each triangle. Round side measures to the nearest tenth and angle measures to the nearest degree.

3. 3

10

r 4.

b

c49°

9

Study Guide and Intervention (continued)

Right Triangle Trigonometry

Example

4-1

2035

a


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Find the exact values of the six trigonometric functions of θ.

1.

3

8

θ

2. 7

24

θ

Find the value of x. Round to the nearest tenth, if necessary.

3.

63.1

56°

x

4.

19.2

26°

x

5. On a college campus, the library is 80 yards due east of the dormitory and the recreation center is due north of the library. The college is constructing a sidewalk from thedormitory to the recreation center. The sidewalk will be at a 56° angle with the current sidewalk between the dormitory and the library. To the nearest yard, how long will the new sidewalk be?

6. If cot A = 8, find the exact values of the remaining trigonometric functions for the acute angle A.

Find the measure of angle θ. Round to the nearest degree, if necessary.

7.

2 3

θ

8.

7

4

θ

Solve each triangle. Round side measures to the nearest tenth and angle measures to the nearest degree.

9.

14

22° 10.

72

c

11. SWIMMING The swimming pool at Perris Hill Plunge is 50 feet long and 25 feet wide. If the bottom of the pool is slanted so that the water depth is 3 feet at the shallow end and 15 feet at the deep end, what is the angle of elevation at the bottom of the pool?

PracticeRight Triangle Trigonometry

4-1


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Word Problem PracticeRight Triangle Trigonometry

1. MONUMENTS The Leaning Tower of Pisa in Italy is about 55.9 meters tall and is leaning so it is only about 55 meters above the ground. At what angle is the tower leaning?

2. SUBMARINES A submarine that is 250 meters below the surface of the ocean begins to ascend at an angle of 22° from vertical. How far will the submarine travel before it breaks the surface of the water?

3. PHYSICS Suppose you are traveling in a car when a beam of light passes from the air to the windshield. The measure of the angle of incidence θi is 55°, and the measure of the angle of refraction θr is 35.25°. Use Snell’s Law, sin θi −

sin θr = n,

to find the index of refraction n of the windshield to the nearest thousandth.

4. CONSTRUCTION A 30-foot ladder leaning against the side of a house makes a 70.1° angle with the ground.

30 ft

70.1°

a. How far up the side of the house does the ladder reach?

b. What is the horizontal distance between the bottom of the ladder and the house?

5. OBSERVATION A person standing 100 feet from the bottom of a cliff notices a tower on top of the cliff. The angle of elevation to the top of the cliff is 30°, and the angle of elevation to the top of the tower is 58°. How tall is the tower?

58°

30°

100 ft

6. GEOMETRY The apothem of a regular polygon is the measure of the line segment from the center of the polygon to the midpoint of one of its sides. A circle is circumscribed about a regular hexagon with an apothem of 4.8 centimeters.

4.8 cm

a. Find the radius of the circumscribed circle.

b. What is the length of a side of the hexagon?

c. What is the perimeter of the hexagon?

7. SKATEBOARD Suppose you want to construct a ramp for skateboarding with a 19° incline and a height of 4 feet.

a. Draw a diagram to represent the situation.

b. Determine the length of the ramp.

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Angles and Their Measures One complete rotation can be represented by 360° or 2π radians. Thus, the following formulas can be used to relate degree and radian measures.

Degree/Radian Coversion Rules

1° = π −

180 radians 1 radian = ( 180

− π

) °

If two angles have the same initial and terminal sides, but different measures, they are called coterminal angles.

Write each degree measure in radians as a multiple of π and each radian measure in degrees.

a. 36°

36° = 36° ( π radians

− 180°

) Multiply by π radians −

180°

.

= π

− 5 radians or π −

5 Simplify.

b. - 17π

− 3

- 17π

− 3 = -

17π −

3 radians Multiply by

180° −

π radians .

= - 17π

− 3 radians (

180° −

π radians

) = -1020° Simplify.

Exercises


1. -250° 2. 6° 3. -145°

4. 870° 5. 18° 6. -820°

7. 4π 8. 13π −

30 9. -1

10. 3π −

16 11. -2.56 12. -

7π −

9

Identify all angles that are coterminal with the given angle.

13. - π

− 2 14. 135° 15. 5π

− 3

Example

Study Guide and InterventionDegrees and Radians

4-2


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Applications with Angle Measure The rate at which an object moves along a circular path is called its linear speed. The rate at which the object rotates about a fixed point is called its angular speed.

Determine the angular speed and linear speed if 8.2 revolutions are completed in 3 seconds and the distance from the center of rotation is 7 centimeters. Round to the nearest tenth.The angle of rotation is 8.2 × 2π or 16.4π radians.

ω = θ − t Angular speed

= 16.4π − 3 θ = 16.4π radians and t = 3 seconds

≈ 17.17403984 Use a calculator.

Therefore, the angular speed is about 17.2 radians per second.

The linear speed is rθ − t .

V = s − t Linear speed

= rθ − t s = rθ

= 7(16.4π)

− 3

r = 7 centimeters, θ = 16.4π radians, and t = 3 seconds

= 120.218278877 Use a calculator.

Therefore, the linear speed is about 120.2 centimeters per second.

ExercisesFind the rotation in revolutions per minute given the angular speed and the radius given the linear speed and the rate of rotation.

1. ω = 2.7 rad/s 2. ω = 4 − 3 π rad/hr

3. ω = 3 − 2 π rad/min 4. V = 24.8 m/s, 120 rev/min

5. V = 118 ft/min , 3.6 rev/s 6. V = 256 in./h, 0.5 rev/min


Degrees and Radians

Example

Suppose an object moves at a constant speed along a circular path of radius r.

If s is the arc length traveled by the object during time t, then the object’s linear speed v is given by

v = s − t .

If θ is the angle of rotation (in radians) through which the object moves

during time t, then the angular speed ω of the object is given by

ω = θ − t .

θ

4-2


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Write each decimal degree measure in DMS form and each DMS measure in decimal degree form to the nearest thousandth.

1. 28.955 2. -57.3278

3. 32 28' 10" 4. -73 14' 35"


5. 25° 6. 130°

7. 3π −

4 8. 5π

− 3

Identify all angles that are coterminal with the given angle. Then find and draw one positive and one negative angle coterminal with the given angle.

9. 43° 10. − 7π

− 4

Find the length of the intercepted arc with the given central angle measure in a circle of the given radius. Round to the nearest tenth.

11. 30°, r = 8 yd 12. 7π − 6 , r = 10 in.

Find the rotation in revolutions per minute given the angular speed and the radius given the linear speed and the rate of rotation.

13. ω = 4 − 5 π rad/s 14. V = 32 m/s, 100 rev/min

15. On a game show, a contestant spins a wheel. The angular speed of the wheel was ω = π − 3 radians per second. If the wheel maintained this rate, what would be the rotation in revolutions per minute?

Find the area of each sector.

16. θ = π − 6 , r = 14 in. 17. θ = 7π

− 4 , r = 4 m

PracticeDegrees and Radians

4-2


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Word Problem PracticeDegrees and Radians

1. TRANSPORTATION A curve on a highway has a 1000-foot radius with lanes that are 14-feet wide. If the curve makes a 180° turn, how much longer is the outside edge of the far lane than the inside edge of the close lane?

90°

1000 ft14 ft

2. OLYMPICS In the 2008 Olympics in Beijing, China, Primoz̃ Kozmus won the gold medal in the hammer throw. If he rotated 195° before releasing the hammer, what other angles would be coterminal with this angle? Describe all possible options and explain what it means for this situation.

3. GARDEN Jayden is planting a circular garden in her backyard that will have a radius of 10 feet, as shown below.

10 ft

a. What area does she want to use for tomatoes?

b. What area does she want to use for wildflowers?

c. If six ounces of wildflower seeds can cover 10 square feet, how many ounces of seeds does Jayden need to buy in order to plant all of the wildflowers?

4. WIND TURBINE One type of wind turbine blade is about 27.1 meters long. If the blades of the turbine are rotating at 28.5 revolutions per minute, what are the angular speed and linear speed to the nearest tenth?

5. CENTRIFUGE A centrifuge is a machine that is used to separate fluids. In a high school lab, the centrifuge has an angular speed of 400π radians per second.

a. Find the rotation in revolutions per minute.

b. What is the linear speed of a substance that is 8 centimeters away from the center?

6. CLOCK A clock is divided into 12 sectors each with an angle of π −

6 radians.

a. How many hours pass when the hour hand moves 77π radians?

b. If the clock’s radius is 14 inches,what is the area of the smallersector formed by the hands at the 5:00 position?

c. What is the angular speed of the hour hand and minute hand?

7. CD PLAYER A CD player spins at about 200 revolutions per minute. Through what angle does a point on the edge of the CD spin in one minute?

4-2


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Trigonometric Functions of Any Angle The definitions of the six trigonometric functions may be extended to include any angle as shown below.

You can use the following steps to find the value of a trigonometric function of any angle θ.1. Find the reference angle θ '.2. Find the value of the trigonometric function for θ '.3. Use the quadrant in which the terminal side of θ lies to determine the sign of

the trigonometric function value of θ.

Let (−9, 12) be a point on the terminal side of an angle θ in standard position. Find the exact values of the six trigonometric functions of θ.

Use the values of x and y to find r.r =

√ �� x 2 + y 2 Pythagorean Theorem

= √ �� (-9) 2 + 12 2 x = -9 and y = 12

= √ �� 225 or 15 Take the positive square root.

Use x = −9, y = 12, and r = 15 to write the six trigonometric ratios.

sin θ = y − r = 12 −

15 or 4 −

5 cos θ = x − r = -9 −

15 or -

3 − 5 tan θ =

y − x = 12 −

-9 or - 4 −

3

csc θ = r − y = 15 − 12

or 5 − 4 sec θ = r − x = 15 −

-9 or -

5 − 3 cot θ = x − y = -9 −

12 or -

3 − 4

Exercises

The given point lies on the terminal side of an angle θ in standard position. Find the values of the six trigonometric functions of θ.

1. (2, −5) 2. (12, 4) 3. (-3, -8)

Example

Study Guide and InterventionTrigonometric Functions on the Unit Circle

4-3

Let θ be any angle in standard position and point P(x, y) be a point on the terminal side of θ.

Let r represent the nonzero distance from P to the origin. That is, let r = √ �� x 2 + y 2 ≠ 0.

Then the trigonometric functions of θ are as follows.

sin θ = y − r csc θ = r − y , y ≠ 0

cos θ = x − r sec θ = r − x , x ≠ 0

tan θ = y − x , x ≠ 0 cot θ = x − y , y ≠ 0

y

x

P(x, y)

x

yr

θ

y

x

θ

(-9, 12)

Find the exact value of each expression.

4. sin 5π −

3 5. csc 210° 6. cot (-315°)


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Trigonometric Functions on the Unit Circle You can use the unit circle to find the values of the six trigonometric functions for θ. The relationships between θ and the point P(x, y) on the unit circle are shown below.

Therefore, the coordinates of P corresponding to the angle t can be written as P(cos t, sin t).

Find the exact value of tan 5π −

3 . If undefined, write undefined.

5π −

3 corresponds to the point (x, y) = ( 1 −

2 , -

√ � 3 −

2 ) on the unit cirle.

tan t = y − x Definition of tan t

tan 5π −

3 =

-

√ � 3 −

2 −

1 − 2 x = 1 −

2 and y = -

√ � 3 −

2 when t = 5π

− 3

tan 5π −

3 = - √ � 3 Simplify.

Exercises

Find the exact value of each expression. If undefined, write undefined.

1. tan π − 2 2. sec −

3π −

4

3. cos 7π −

6 4. sin 5π

− 4

5. cot 4π −

3

6. csc −

5π −

3

7. tan −60° 8. cot 270°


Trigonometric Functions on the Unit Circle

Example

4-3

Let t be any real number on a number line and let P(x, y) be the point

on t when the number line is wrapped onto the unit circle. Then the

trigonometric functions of t are as follows.

sin t = y cos t = x tan t = y − x , x ≠ 0

csc t = 1 − y , y ≠ 0 sec t = 1 − x , x ≠ 0 cot t = x − y , y ≠ 0

y

x

t

t

P(x, y) orP(cos t, sin t)

1


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The given point lies on the terminal side of an angle θ in standard position. Find the values of the six trigonometric functions of θ.

1. (−1, 5) 2. (7, 0) 3. (−3, −4)

4. (1, −2) 5. (−3, 1) 6. (2, −4)

Sketch each angle. Then find its reference angle.

7. 330° 8. − 3π

− 4 9. 7π

− 6

10. 7π −

4 11. 135° 12. −

π −

3

Find the exact value of each expression. If undefined, write undefined.

13. csc 90° 14. tan 270° 15. sin (−90°)

16. cos 3π −

2 17. sec (−

π −

4 ) 18. cot 5π

− 6

19. PENDULUMS The angle made by the swing of a pendulum and its vertical resting position can be modeled by θ = 3 cos πt, where t is time measured in seconds and θ is measured in radians. What is the angle made by the pendulum after 6 seconds?

PracticeTrigonometric Functions on the Unit Circle

4-3


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1. NAVIGATION A ship’s course is plotted on a coordinate grid with the origin of the grid representing the port from which the ship departed. When the captain of the ship radios to the port, he is told that the coordinates of his location are (13, −8).

a. At what angle did the ship travel with respect to due east?

b. How far has the ship traveled? Round to the nearest tenth.

2. CONDITIONING The Dot Drill is a conditioning drill that is used in some athletic programs. The drill uses 5 dots that are placed on the floor as shown. The athletes jump in different sequences as fast as they can. The center dot is placed half-way between the corner dots.

3 ft

2 ft

1

3

2

54

a. How far away is the middle dot from the corner dot?

b. Suppose an athlete standing on the center dot facing dot 2 turns clockwise until she faces dot 1. Find her angle of rotation.

c. If an athlete jumps from dots 1 to 2 to 3 to 4 to 5, how many feet has the athlete jumped?

3. HOCKEY The goals in a hockey rink are 6 feet wide. A hockey player is standing 25 feet directly in front of the left side of one of the goals.

a. What is the maximum angle at which the player can shoot the puck to make it in the goal?

b. If the player skates forward 5 feet, how wide of a range of angles can the player shoot the puck to make it in the goal?

c. Describe what happens to the maximum angle as the player gets closer to the goal.

4. KITES Suppose the string of a kite makes a 78° angle with the ground. If 45 feet of string is released, how high off the groundis the kite?

5. CURRENT The current I in amperes for a circuit at time t seconds is given by I = 40 cos [60π(t +

1−120)].

a. Find the maximum current.

b. Find the current after 0.02 second.

6. WINDSHIELD WIPER A 12-inch windshield wiper blade rotates 120° as shown. Find the coordinates of the starting and ending points A and B of the tip of the blade relative to the pivot point O.

30° 30°12120°

Word Problem PracticeTrigonometric Functions on the Unit Circle

4-3


PreCal-B

Student Name: __________________

Student ID: _____________________

School Name: ___________________

Summer School 2020

Week 2

Teacher Name: __________________

High School Math

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Transformations of Sine and Cosine Functions A sinusoid is a transformation of the graph of the sine function. The general form of the sinusoidal functions sine and cosine are y = a sin (bx + c) + d or y = a cos (bx + c) + d. The graphs of y = a sin (bx + c) + d and y = a cos (bx + c) + d have the following characteristics.

Amplitude = |a| Period = 2π

−

|b|

Frequency = |b|

− 2π

or 1 − period

Phase Shift = - c −

|b|

Vertical Shift = d Midline y = d

State the amplitude, period, frequency, phase shift, and vertical shift of y = -2 cos (

x − 4 - π −

3 ) + 2. Then graph two periods of the function.

Amplitude = |a| = |-2| = 2

Period = 2π −

|b|

= 2π −

⎪ 1 − 4 ⎥ or 8π

Frequency = |b|

− 2π

= ⎪ 1 − 4 ⎥ −

2π

or 1 − 8π

Phase Shift =- c −

|b|

= - (-

π −

3 ) −

⎪ 1 − 4 ⎥ or 4π

− 3

Vertical Shift = d or 2

Exercises

State the amplitude, period, frequency, phase shift, and vertical shift of each function. Then graph two periods of the function.

1. y = 3 sin (2x + π) 2. y = cos (x - π − 3 ) + 2

x

y4

−4

−2π

2π 3π

2

x

y

2

4

−4

−24π

37π

310π

3

State the frequency and midline of each function.

3. y = 3 sin ( 1 − 2 x + 4π

− 3 ) + 1 4. y = cos (3x) - 2

Write a sine function with the given characteristics.

Study Guide and InterventionGraphing Sine and Cosine Functions

Example

x

y

1

4

4π

316π

328π

340π

3

4-4

5. amplitude = 2, period = 4, phase shift = 1 − 2 , vertical shift = 4

6. amplitude = 1.2, phase shift = 3π −

2 , vertical shift = 1


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Applications of Sinusoidal Functions You can use sinusoidal functions to solve certain application problems.

The table shows the average monthly temperatures for Ann Arbor, Michigan, in degrees Fahrenheit (°F). Write a sinusoidal function that models the average monthly temperatures as a function of time x, where x = 1 represents January.The data can be modeled by a sinusoidal function of the form y = a sin (bx + c) + d. Find the maximum M and minimum m values of the data, and use these values to find a, b, c, and d.a = 1 −

2 (M − m) Amplitude formula

= 1 − 2 (84 − 30) or 27 M = 84 and m = 30

d = 1 − 2 (M + m) Vertical shift formula

= 1 − 2 (84 + 30) or 57 M = 84 and m = 30

Period = 2 (xmax - xmin) xmax

= July or month 7 and

= 2(7 − 1) or 12 xmin

= January or month 1

⎪b⎥ = 2π −

12 or π −

6 ⎪b⎥ = 2π

− period

Phase shift = − c −

⎪b⎥ Phase shift formula

4 = − c −

π − 6 Phase shift = 4 and ⎪b⎥ = π −

6

c = − 2π −

3 Solve for c.

Therefore, y = 27 sin ( π − 6 x − 2π

− 3 ) + 57 is one model for the average

monthly temperature in Ann Arbor, Michigan.

Exercise

1. MUSEUM ATTENDANCE The table gives the number of visitors in thousands to a museum for each month.

a. Write a trigonometric function that models the monthly attendance at the museum using x = 1 to represent January.

b. According to your model, how many people should the museum expect to visit during October?


Graphing Sine and Cosine Functions

Example

4-4

Month Temperature

Jan 30°

Feb 34°

Mar 45°

Apr 59°

May 71°

June 80°

July 84°

Aug 81°

Sept 74°

Oct 62°

Nov 48°

Dec 35°

Month Jan Feb Mar Apr May Jun Jul Aug

Visitors 10 8 11 15 24 30 32 29


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Describe how the graphs of f(x) and g(x) are related. Then find the amplitude of g(x) and sketch two periods of both functions on the same coordinate axes.

1. f(x) = sin x

g(x) = 1 − 3 sin x

2. f(x) = cos x

g(x) = - 1 − 4 cos x

State the amplitude, period, frequency, phase shift, and vertical shift of each function. Then graph two periods of the function.

3. y = 2 sin (x + π − 2 ) - 3 4. y = 1 −

2 cos (2x − π) + 2

Write a sinusoidal function with the given amplitude, period, phase shift, and vertical shift.

5. sine function: amplitude = 15, period = 4π, phase shift = π − 2 , vertical shift = -10

6. cosine function: amplitude = 2 − 3 , period = π −

3 , phase shift = -

π −

3 , vertical shift = 5

7. MUSIC A piano tuner strikes a tuning fork note A above middle C and sets in motion vibrations that can be modeled by y = 0.001 sin 880tπ. Find the amplitude and period of

the function.

PracticeGraphing Sine and Cosine Functions

y

xπ 2π 4π

-1

1y

xπ 2π 3π 4π

-1

1

y

x

2

4

-4

-2π 2π-π π

y

x

2

4

-2

-4

-π ππ

2π

2-

4-4


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1. METEOROLOGY The average monthly temperatures for Baltimore, Maryland, are shown below.

Month Temperature (°F) Month Temperature (°F)

Jan 32 July 77

Feb 35 Aug 76

Mar 44 Sept 69

Apr 53 Oct 57

May 63 Nov 47

June 73 Dec 37

a. Determine the amplitude, period, phase shift, and vertical shift of a sinusoidal function that models the monthly temperatures using x = 1 to represent January.

b. Write an equation of a sinusoidal function that models the monthly temperatures.

c. According to your model, what is Baltimore’s average temperature in July? December?

2. BOATING A buoy, bobbing up and down in the water as waves pass it, moves from its highest point to its lowest point and back to its highest point every 10 seconds. The distance between its highest and lowest points is 3 feet.

a. Determine the amplitude and period of a sinusoidal function that models the bobbing buoy.

b. Write an equation of a sinusoidal function that models the bobbing buoy, using x = 0 as its highest point.

3. A student graphed a periodic function with a period of n. The student then translated the graph c units to the right and obtained the original graph. Describe the relationship between c and n.

4. SWING Marsha is pushing her brother Bobby on a rope swing over a creek. When she starts the swing, he is 7 feet over land away from the edge of the creek. After 2 seconds, Bobby is 11 feet over the water past the edge of the creek. Assume that the distance from the edge of the creek varies sinusoidally with time and that the distance y is positive when Bobby is over the water and negative when he is over land. Write a trigonometric function that models the distance Bobby is from the edge of the creek at time t seconds.

5. ROLLER COASTER Part of a roller coaster track is a sinusoidal function. The high and low points are separated by 150 feet horizontally and 82 feet vertically as shown. The low point is 6 feet above the ground.

150 ft6 ft

82 ft

a. Write a sinusoidal function that models the distance the roller coaster track is above the ground at a given horizontal distance x.

b. Point A is 40 feet to the right of the y-axis. How far above the ground is the track at point A?

Word Problem PracticeGraphing Sine and Cosine Functions

4-4


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Tangent and Reciprocal Functions You can use the same techniques you learned for graphing the sine and cosine functions to graph the tangent function and the reciprocal trigonometric functions—cotangent, secant, and cosecant. The general form of the tangent function, which is similar to that of the sinusoidal functions, is

y = a tan (bx + c) + d,where a produces a vertical stretch or compression, b affects the period, c produces a horizontal translation, and d produces a vertical shift. The term amplitude does not apply to the tangent or cotangent functions because the heights of these functions are infinite.

Locate the vertical asymptotes and sketch the graph of y = sec (2x + π) + 4.

The period of y = sec (2x + π) + 4 is 2π −

⎪b⎥ or π. Since c = π, the phase shift is −

π −

2 . Therefore,

the vertical asymptotes are located every π − 2 units at -

3π −

4 , - π −

4 , π −

4 , 3π

− 4 , and 5π

− 4 . The vertical

shift is 4. Create a table including the relative maximum and minimum points for the period

[- π − 2 , 3π −

2 ].

FunctionsVertical

Asymptote

Relative

Minimum

Vertical

Asymptote

Relative

Maximum

Vertical

Asymptote

Relative

Minimum

Vertical

Asymptote

y = sec x x = - π − 2 (0, 1) x = π −

2 (π, -1) x = 3π

− 2 (2π, 1) x = 5π

− 2

y = sec (2x) x = - π − 4 (0, 1) x = π −

4 ( π −

2 , -1) x = 3π

− 4 (π, 1) x = 5π

− 4

y = sec (2x + π) x = - 3π −

4 (- π −

2 , 1) x = - π −

4 (0, -1) x = π −

4 ( π −

2 , 1) x = 3π

− 4

y = sec (2x + π) + 4 x = - 3π −

4 (-

π −

2 , 5) x = - π −

4 (0, 3) x = π −

4 ( π −

2 , 5) x = 3π

− 4

Graph one cycle on the interval [- π − 2 , 3π

− 2 ].

Then sketch one cycle to the left and right.

ExercisesLocate the vertical asymptotes and sketch the graph of each function.

1. y = 3 tan (4πx) 2. y = csc (2x - 3π)

Study Guide and InterventionGraphing Other Trigonometric Functions

Example

y

-π πθ

2

6

8

10

-2-π

2π

2

4-5

x14

14

12

-12

-x

y

3π

2π

23π

2-π

2-


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Damped Trigonometric Functions A damped trigonometric function is the product of a sinusoidal function of the form y = sin bx or y = cos bx and another function y = f(x), called the damping factor. The graph of the product function is the oscillation of the sinusoidal function between the graphs of y = f(x) and y = -f(x). The resulting graph is called a damped wave, and the reduction in the amplitude is known as damped oscillation. Damped oscillation can occur as x approaches ±∞, as x approaches 0, or both.

Damped harmonic motion occurs when the amplitude of a function is damped due to friction over time. An object is in damped harmonic motion when the amplitude is determined by the function a(t) = ke –ct. For y = ke –ct sin ωt and y = ke –ct cos ωt, where c > 0, k is the displacement, c is the damping constant, t is time, and ω is the period.

Identify the damping factor f(x) of y = 1 − 2 x2 sin 5x. Then use a

graphing calculator to sketch the graphs of f(x), −f(x), and the given function in the same viewing window. Describe the behavior of the graph.

The function y = 1 − 2 x2 sin 5x is the product of the functions y = 1 −

2 x2

and y = sin 5x. Therefore, the damping factor is f(x) = 1 − 2 x2.

The amplitude of the function is decreasing as x approaches zero.

ExercisesIdentify the damping factor f(x) of each function. Then use a graphing calculator to sketch the graphs of f(x), −f(x), and the given function in the same viewing window. Describe the behavior of the graph.

1. y = 3x sin 2x 2. y = 1 − 2 x cos 4x

3. A guitar string is plucked at a distance of 0.9 centimeter above its resting position and then released, causing vibration. The damping constant of the guitar string is 1.8, and the note produced has a frequency of 185 cycles per second.

a. Write a trigonometric function that models the motion of the string.

b. Determine the amount of time t that it takes the string to be damped so that -0.28 ≤ y ≤ 0.28.


Graphing Other Trigonometric Functions

Example

[-2π, 2π] scl: by [-8, 8] scl: 12π

4-5


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Locate the vertical asymptotes, and sketch the graph of each function.

1. y = −3 tan x 2. y = −2 cot (2x + π − 3 )

x

y

-π π-2π

-2

2

2π

x

y

-π π-2π

-2

2π

2

3. y = csc x + 3 4. y = sec ( x − 3 + π) – 1

-1-2

2345678

θ

y

-3π

23π

2

1

-1

-3-4

234

-3π 3π θ

y

Identify the damping factor f(x) of the function. Then use a graphing calculator to sketch the graphs of f(x), −f(x), and the given function in the same viewing window. Describe the behavior of the graph.

5. y = 1 − 2 x cos 2x 6. y = −

3 − 2 x sin πx −

2

7. MUSIC A guitar string is plucked at a distance of 0.6 centimeter above its resting position and then released, causing vibration. The damping constant of the guitar string is 1.8, and the note produced has a frequency of 105 cycles per second.


b. Determine the amount of time t that it takes the string to be damped so that -0.24≤ y ≤ 0.24.

PracticeGraphing Other Trigonometric Functions

4-5


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1. EYESIGHT The observation deck of the Space Needle in Seattle, Washington, is 526 feet above the ground. A six-foot-tall man is watching a car on the street below. Let d represent the distance from the man to the car and θ the angle of depression. Write d as a function of θ.

2. AMUSEMENT PARKS The SpaceShot is a ride at an amusement park that propels passengers straight into the air. A spectator is standing 20 meters away from the ride. Let h be the height of the passengers on the ride from their original starting position and θ be the angle of elevation to the passengers from the spectator.

θ

h

20 m

a. Write h as a function of θ.

b. Graph the function on the interval 0° ≤ θ < 90°.

h

90°45°

20

40

θ

c. Approximate how high the passengers are when the angle of elevation is 55°.

3. MUSIC A banjo string is plucked at a distance of 0.3 centimeter above its resting position and then released, causing vibration. The damping constant of the banjo string is 2.3, and the note produced has a frequency of 180 cycles per second.


b. Determine the amount of time t that it takes the string to be damped so that −0.1 ≤ y ≤ 0.1.

4. ANT José is watching an ant climb up the downspout. He is standing 2 meters from the downspout. Let d be the distance the ant is up the downspout from a point even with José’s eyes and θ be the angle of elevation to the ant from José.

d

2 mθ

a. Write d as a function of θ.

b. Graph the function on the interval 0° ≤ θ < 90°.

c. About how far up the downspout is the ant when the angle of elevation is 72°?

Word Problem PracticeGraphing Other Trigonometric Functions

4-5

d

90°

2

4

θ


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Inverse Trigonometric Functions When restricted to a certain domain, the sine, cosine, and tangent functions have inverse functions known as the arcsine, arccosine, and arctangent functions, respectively.

Inverse Trigonometric Functions

Inverse Sine of xy = sin -1 x or y = arcsin x if and only if

sin y = x, for -1 ≤ x ≤ 1 and - π − 2 ≤ y ≤ π −

2 .

Inverse Cosine of xy = cos -1 x or y = arccos x if and only if

cos y = x, for -1 ≤ x ≤ 1 and 0 ≤ y ≤ π.

Inverse Tangent of xy = tan -1 x or y = arctan x if and only if

tan y = x, for -∞ ≤ x ≤ ∞ and - π − 2 < y < π −

2 .

Find the exact value of each expression, if it exists.

1. sin -1 (- 1 − 2 )

Find a point on the unit circle in the interval

[ - π

− 2 , π −

2 ] with a y-coordinate of −

1 − 2 . When

t = − π

− 6 , sin t = − 1 −

2 . Therefore, sin -1 (- 1 −

2 ) = − π −

6 .

2. cos -1 4

Because the domain of the inverse cosine function is [−1, 1] and 4 > 1, there is no angle with a cosine of 4. Therefore, the value of cos -1 4 does not exist.

Exercises


1. arctan 0 2. arcsin √ 3 −

2

3. cos −1 √ 2 −

2

4. tan −1 (−1)

Sketch the graph of the function.

5. y = arcsin 2x – 1 6. y = tan −1 (x – 1)

y

x-2 2-4 4π

2

-π

-

π

2

π

y

x-

-1 1π

2

-π

-

π

2

12

12

Study Guide and InterventionInverse Trigonometric Functions

4-6

Examples

y

x

√32

12

-,

π

6t = -

- 1

1

1- 1

⎛⎝

⎞⎠


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Compositions of Trigonometric Functions Because the domains of the trigonometric functions are restricted to obtain the inverse trigonometric functions, the composition of a trigonometric function and its inverse does not follow the rules that you learned in Lesson 1-7. The properties that apply to trigonometric functions and their inverses are summarized below.

Inverse Properties of Trigonometric Functions

f( f -1 (x)) = x f -1 (f(x)) = x

If –1 ≤ x ≤ 1, then sin ( sin –1 x) = x. If – π − 2 ≤ x ≤ π −

2 , then sin –1 (sin x) = x.

If –1 ≤ x ≤ 1, then cos ( cos –1 x) = x. If 0 ≤ x ≤ π, then cos –1 (cos x) = x.

If –∞ < x < ∞, then tan ( tan –1 x) = x. If – π − 2 < x < π −

2 , then tan –1 (tan x) = x.

Find the exact value of cos [ tan -1 - 4 − 3 ] .

To simplify the expression, let u = tan –1 (- 4 − 3 ) , so tan u = – 4 −

3 .

Because the tangent function is negative in Quadrants II and IV and

the domain of the inverse tangent function is restricted to Quadrants

I and IV, u must lie in Quadrant IV.

Using the Pythagorean Theorem, you can find that the length of the

hypotenuse is 5. Now solve for cos u.

cos u = adj

− hyp

Cosine function

= 3 − 5 adj = 3 and hyp = 5

So, cos [ tan -1 - 4 − 3 ] = 3 −

5 .

Exercises


1. sin ( sin −1 − 3 − 4 ) 2. cos −1 (cos π −

2 )

3. tan −1 (tan 3π −

2 ) 4. sin −1 (cos π −

6 )

5. cos (arcsin 1 − 2 )

6. tan (arcsin −

1 − 2 )

7. cos (arccos 2) 8. cos (arctan −

√ � 3 −

3 )

4-6 Study Guide and Intervention (continued)

Inverse Trigonometric Functions

Example

y

-4

3xu


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Sketch the graph of each function.

1. y = arccos 3x 2. y = arctan x + 1

y

x-0.5 0.5-1 1π

2

-π

π

-

π

2

y

x-1 1-2 2π

2

-π

π

-

π

2

3. y = sin −1 3x 4. y = tan −1 3x

y

x-0.5 0.5-1 1π

2

-π

π

-

π

2

y

x-1 1-2 2π

2

-π

π

-

π

2


5. arcsin −1 (-

√ � 3 −

2 ) 6. cos −1 (cos π −

3 )

7. tan (− 3π

− 2 ) 8. sin −1 (cos π −

3 )

9. arctan (-

√ � 3 −

3 ) 10. arcsin (−

1 − 2 )

11. tan ( sin -1 1 - cos -1 1 − 2 )

12. sin (arctan −

√ � 3 −

3 )

13. ART Hans purchased a painting that is 30 inches tall that will hang 8 inches above the fireplace. The top of the fireplace is 55 inches from the floor.

a. Write a function modeling the maximum viewing angle θ for the distance d for Hans if his eye-level when sitting is 2.5 feet above the ground.

b. Determine the distance that corresponds to the maximum viewing angle.

Write each trigonometric expression as an algebraic expression of x.

14. sin (arccos x) 15. tan ( sin −1 x)

4-6 PracticeInverse Trigonometric Functions


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4-6 Word Problem PracticeInverse Trigonometric Functions

1. BUSINESS A retail company is installing a ramp outside its building to make it easier to roll merchandise on carts. If the bottom of the ramp is going to be 22 feet from the base of the door and the ramp needs to go down 4 feet, what is the measure of the angle the ramp makes with the ground?

22 ft

4 ftθ

2. PROJECTION A teacher is projecting a graph from an overhead projector onto screen in a lecture hall as shown.

θ

3.5 ft

d

6 ft

6 ft

a. Write a function modeling the maximum projecting angle θ in terms of d.

b. Use a graphing calculator to determine the distance for the maximum projecting angle.

3. ROADS The grade of a road refers to the slope of the incline. If a road is said to have a 3% grade, it means that for each 100 horizontal feet the road rises three feet. A road in the Rocky Mountains has a grade of 6%. What is the angle of the incline?

4. VIDEO At a swim meet, a parent is videotaping his son from a seat in the stands that is 20 meters past the starting line and 8 meters away from his son’s lane as shown. Let x represent the distance the son has swam.

a. Write x as a function of θ.

b. What angle does the parent have the camera at when the race is just starting?

c. What angle does the parent have the camera at when the son has swam 25 meters?

d. Explain the differences in your answers from parts b and c.

5. ZIP-LINE At a park, there is a zip-line that takes riders from a platform that is 25 feet high to a platform that is 2 feet high. If the bases of the platforms are 40 feet apart, what is the measure of the angle the zip-line makes with the lower platform?

6. RAMP A wheelchair ramp leading up to a porch is shown. Find θ.

24 ft 3 ft

θ

m

8 mθ

20 m


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Solve Oblique Triangles The Law of Sines can be used to solve an oblique triangle when given the measures of two angles and a nonincluded side (AAS), two angles and the included side (ASA), or two sides and a nonincluded angle (SSA).

The Law of Cosines can be used to solve an oblique triangle when given the measures of three sides (SSS) or the measures of two sides and their included angle (SAS).

Solve � ABC. Round side lengths to the nearest tenth and angle measures to the nearest degree.

Because two angles are given, C = 180° - (100° + 30°) or 50°.

Use the Law of Sines to find b and c.

sin A − a = sin B − b Law of Sines sin A − a = sin C − c

sin 30° −

15 = sin 100°

− b Substitution sin 30°

− 15

= sin 50° − c

b sin 30° = 15 sin 100° Cross products c sin 30° = 15 sin 50°

b = 15 sin 100° −

sin 30°

Divide each side by sin 30°. c = 15 sin 50° −

sin 30°

b ≈ 29.5 Use a calculator. c ≈ 23.0

Therefore, b ≈ 29.5, c ≈ 23.0, and C = 50°.

Exercises

Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree.

1. 1220

28

2.

8

10

100°

3.

150°

20020°

4.

45°

30°10

4-7 Study Guide and InterventionThe Law of Sines and the Law of Cosines

Example


15100°

30°

c

b


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Find Areas of Oblique Triangles When the measures of all three sides of an oblique triangle are known, Heron’s Formula can be used to find the area of the triangle.

Heron’s Formula

If the measures of the sides of � ABC are a, b, and c,

then the area of the triangle is

√ �� s(s - a)(s - b) (s - c) ,

where s = 1 − 2 (a + b + c).

When two sides and the included angle of a triangle are known, the area is one-half the product of the lengths of the two sides and the sine of the included angle.

Area of a Triangle Given SAS

Area = 1 − 2 bc sin A

Area = 1 − 2 ac sin B

Area = 1 − 2 ab sin C

Find the area of � XYZ to the nearest tenth.

The value of s is 1 − 2 (20 + 22 + 36) or 39.

Area = √ �� s(s - x)(s - y)(s - z) Heron’s Formula

= √ �� 39(39 - 20)(39 - 22)(39 - 36) s = 39, x = 20,

y = 22, and z = 36

= √ �� 37,791 or about 194.4 in2 Simplify.

Exercises

Use Heron’s Formula to find the area of each triangle. Round to the nearest tenth.

1. � ABC if a = 14 ft, b = 9 ft, c = 8 ft 2. � FGH if f = 8 in., g = 9 in., h = 3 in.

3. � MNP if m = 3 yd, n = 4.6 yd, p = 5 yd 4. � XYZ if x = 8 cm, y = 12 cm, z = 13 cm

Find the area of each triangle to the nearest tenth.

5. � RST if R = 50°, s = 12 yd, t = 14 yd 6. � MNP if n = 14 ft, P = 110°, N = 25°

7. � DEF if d = 15 ft, E = 135°, f = 18 ft 8. � JKL if j = 4.3 m, l = 3.9 m, K = 82°

4-7 Study Guide and Intervention (continued)

The Law of Sines and the Law of Cosines

Example


a

c

b

20 in.36 in.

22 in.

a

b

c


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Solve each triangle. Round to the nearest tenth if necessary.

1.

1538°

63°

2.

29° 33°

41

3.

53°

40

49

4.

57

10

5. STREET LIGHTING A lamp post tilts toward the Sun at a 2° angle from the vertical and casts a 25-foot shadow. The angle from the tip of the shadow to the top of the lamp post is 45°. Find the length of the lamp post.

Use Heron’s Formula to find the area of each triangle. Round to the nearest tenth.

6. � ABC if a = 5 ft, b = 12 ft, c = 13 ft 7. � FGH if f = 11 in., g = 13 in., h = 16 in.

8. � MNP if m = 8 yd, n = 3.6 yd, p = 5.2 yd 9. � XYZ if x = 12 cm, y = 10 cm, z = 15.8 cm

Find the area of each triangle to the nearest tenth.

10. � RST if R = 115°, s = 15 yd, t = 20 yd 11. � MNP if n = 4 ft, P = 69°, N = 37°

12. � DEF if d = 2 ft, E = 85°, F = 19° 13. � JKL if j = 68 cm, l = 110 cm, K = 42.5°

4-7 PracticeThe Law of Sines and the Law of Cosines


25 ft

45°

2°


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1. ART Local artists are painting barn quilts at a Community Center. The quilts are each 8 feet by 8 feet. The symmetrical pattern for a particular barn quilt is shown below.

blue

green

red

yellow6.5 ft

45°

100°

a. What is the area of the yellow triangle?

b. It takes three coats of paint to make a barn quilt. If it takes 3.5 ounces of paint to cover a square foot, how much paint will be needed for the yellow triangle?

2. ARCHITECTURE The roof on a househas one side that is in the shape of an isosceles triangle. If the sides of this part are 18 feet long and the angle atthe peak is 50°, what is the area of this part of the roof?

3. ORIENTEERING During an orienteering exercise, two hikers start at point A and head in a direction 30° west of south to point B. They hike 6 miles from point A to point B. From point B, they hike to point C and then from point C back to point A, which is 8 miles directly north of point C. How many miles did they hike from point B to point C?

4. BLIMP A blimp hovers over a soccer stadium. Players 500 feet apart at opposite ends of the stadium with the blimp between them measure the respective angles of elevation to the blimp to be 63° and 72°. How high is the blimp?

500 ft

72°63°

5. AVIATION Due to weather conditions, an airplane flies in different directionsas shown in the diagram.

145 mi

80 mi

110°

DestinationCity

Start

a. How far is the airport from the destination city if the direct route is taken?

b. What are the measures of the two other angles in the triangle?

6. PROPERTY MAINTENANCE The McSweeneys plan to fence a triangular parcel of their land. One side of the property is 75 feet in length and forms a 38° angle with another side of the property, which has not yet been measured. The remaining side of the property is 95 feet in length. Approximate to the nearest tenth the length of fence needed to enclose this parcel of the McSweeneys’ lot.

7. SHIP A ship at sea is 92 miles from one radio tower and 124 miles from another. The angle between the radio signals has a measure of 156°. Find the distance between the radio towers.

4-7 Word Problem PracticeThe Law of Sines and the Law of Cosines

30°

8 mi

6 mi


PreCal-B

Student Name: __________________

Student ID: _____________________

School Name: ___________________

Summer School 2020

Week 3

Teacher Name: __________________

High School Math

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