study notes 1 – right triangle trigonometry (section...
TRANSCRIPT
Unit 4 – Trigonometry
Study Notes 1 – Right Triangle Trigonometry (Section 8.1) Objective: Evaluate trigonometric functions of acute angles. Use a calculator to evaluate trigonometric functions. Use trigonometric functions to model and solve real‐life problems. Identify situations involving an angle of elevation and an angle of depression. By the End of Class I Should… Mini Lesson on Rationalizing the Denominator:
There is an agreement in mathematics that we don’t leave a radical in the denominator of a fraction. To remove the radical from the denominator we multiply the numerator and the denominator by the radical. DON’T FORGET TO SIMPLIFY! Example 1: Simplify the following expressions.
A. √ B.
√ C.
√
Trigonometry means “measurement of angles.”
The easiest angles to deal with in trigonometry are the angles in right triangles. A right triangle is a triangle which has one right angle (90∘) and two acute angles (less than 90∘ each). In the right triangle below, one acute angle is named (theta), and with respect to that angle, you need to be able to identify the opposite side, the adjacent side, and the hypotenuse of the triangle. Basic Right Triangle Trigonometry:
The basic right triangle trigonometric ratios are given by Primary Functions Reciprocal Functions
sine sin
( ) cosecant csc
=
cosine cos
( ) secant sec
=
tangent tan
( ) cotangent cot
=
When using right triangle trigonometry, angles are usually measured in degrees. Example 2: Use the triangle below to find the exact values of the six trigonometric functions of .
sin cos tan
csc sec cot
13
12
5
There are two types of special right triangles for which you should memorize the pattern of side
lengths, this will help build trig ratios. These are 45° 45° 90° and 30° 60° 90° triangles. Example 3: Fill in the sides of the triangles below. Then find the trig ratios for each acute angle.
sin 45∘ cos 45∘ tan 45∘
sin 60∘ cos 60∘ tan 60∘
sin 30∘ cos 30∘ tan 30∘
It is possible to find exact values of trig ratios when using special triangles or right triangles with two given sides. However, approximate values of trig ratios may be needed to solve real‐world problems. Your calculator will find approximate trig ratios in decimal form. Make sure that you have your calculator set in the correct mode. (Degree or Radian) (We will learn more about radians later).
Example 4: Find the following trig ratios using your calculator. (Make sure that you are in degree mode). A. sin 57∘ B. cos 24∘ C. tan 71∘
D. csc 40∘ E. sec 12∘ F. cot 63∘ Example 5: Solve for the missing part labeled for each triangle. A. B. Think About It: What is the difference in solving for a side versus solving for an angle?
40
20°
45
45 60
30
x 6
22
Example 6: If tan , find the following values.
A. cot B. sin C. sec Example 7: Given the point 4, 6 is on the terminal side of an angle in standard position. Determine the exact values of the cosine, cosecant, and cotangent trigonometric functions of the angle.
An angle of elevation is an angle measured
from a horizontal upward toward an object.
An angle of depression is an angle
measured from a horizontal downward toward an object.
Example 8: A biologist wants to know the width (w) of a river in order to properly set instruments for studying the pollutants in the water. From Point A, the biologist walks downstream 70 feet and sights to point C. From
this sighting, it is determined that =54 . How wide is the river?
A 70 ft. Biologist Example 9: A sonar operator on a ship detects a submarine at a distance of 500 meters from the ship and an angle of depression of 40°. How deep is the submarine?
Angle of Depression
Observer
Object
54 =
C
Hint: If you are ever not sure where
to start DRAW A TRIANGLE!!
Angle of Elevation
Observer
Object
Study Notes 2 – Trigonometric Functions of any Angle & the Unit Circle (Section 8.2 & 8.3)
Objective: Use the unit circle to evaluate trigonometric functions. Use a calculator to evaluate trigonometric function for any angle. Find co‐terminal angles and their corresponding trig ratios. By the End of Class I Should…
The previous notes dealt with trigonometric functions for acute angles. This section extends the trigonometric functions to any angle by using reference angles and reference triangles. A discussion of angles (and their measures) in the coordinate plane is an important prerequisite to finding trig ratios for all possible angles. An angle is said to be in standard position in the coordinate plane if it is formed by a rotation from the
positive x‐axis.
Initial side:
Terminal side:
Positive angles:
Negative angles:
The reference angle for an angle in standard position is the acute angle formed by its terminal side and
the horizontal axis (x‐axis)
Example 1: Sketch angles in standard position having the following measures, and find their reference angles.
A. 120° B. 315° C. 150° Reference Angle: Reference Angle: Reference Angle:
x
y
x
y
x
y
x
y
The Unit Circle is the circle centered at the origin with radius 1 unit (hence, the “unit” circle). The equation of this circle is 1. A diagram of the unit circle is shown. We are going to deal primarily with special angles around the unit circle, namely the multiples of 30°, 45°, 60°, and90°. All angles throughout this unit will be drawn in standard position.
Label Coordinates o Quadrant Angles
Label Angles
o Multiples of 90, Multiples of 45, Multiples of 60, Multiples of 30 Unit Circle
Labeling Coordinates for 30, 45, and 60. In standard position sketch a 45 reference angle, and construct a triangle based on the reference angle. Label the coordinates for the point , on the circle that was created by the reference angle of 45. Then repeat the process for 30 and 60.
Label the 1st quadrant of the unit circle with the coordinate points , for 45, 30, and 60. Now using our knowledge of the coordinate plane, fill in the coordinates for quadrants, II, III, IV. Let’s go back to are six trigonometric functions and see how they relate to the unit circle.
Example 2: Find the exact values of the following trigonometric functions: A. Find sin 60° B. cos 90° C. sin 45° D. tan 30° E. csc 135° F. cos 270° G. cos 45° H. sec 150° I. sin 300° J. tan 60° K. csc 150°) M. cot 120° N. cot 225° O. sin 330° P. csc 180° Q. cos 240°) R. tan 180° S. csc 270° T. sin 90°
When angles are not available on the unit circle, use a calculator to evaluate the trigonometric function. Example 3: Use a calculator to find the following:
A. sin 237° B. cos 612° C. csc 112° D. cot 305°
If angles in standard position share the same terminal side, they are called co-terminal angles. The angles in the diagram below have measures of 30°, 390°, and 330°. They are co‐terminal.
Co‐terminal angles have measures which differ by multiples of 360°. Thus, angles which are coterminal to a given angle can be found by adding or subtracting 360° as many times as desired. Example 4: Sketch each angle in standard position, and find one negative and one positive co‐terminal angle for each.
A. 120° B. 405° Since co‐terminal angles share the same terminal side, they form the same reference angles (and reference triangles). Thus, they have the same trig ratios. Example 5: Find the exact values of the following trigonometric functions: A. sin 420° B. cos 585 C. tan 390 D. csc 660 E. sec 450 F. cot 480
Example 6: Given that sin 31° .5150, sin 51° .7771, and sin 71° .9455 (to 4 decimal place accuracy), find the following without using a calculator.
A. sin 431° B. sin 329° C. sin 771°
x
y
x
y
30
390330
Study Notes 3 – Radian Measure for Angles (Section 8.4) Objective: Use radian measure for angles. Convert between radian and degree measure. By the End of Class I Should… We have usually learned to measure an angle in degrees. However, there are other units of
angle measurement. One way that we are going to look at is radians. In many scientific
and engineering calculations, radians are used in preference to degrees. The arc shown has a length chosen equal to the radius; the angle is then 1 radian. By extension an angle of 2 radians will be subtended by an arc of length 2r. Notice that the length of the arc is always given by:
In the general case, the arc length s, is found by
So for a full circle the arc length is the same as its circumference, 2 . Thus,
2 2
In other words, when we are working in radians, the angle in a full circle is 2 radians, thus
° This enables us to have a set of equivalences between degrees and radians.
360° 2
180°
90°2
45°4
60°3
30°6
**Finishing Labeling the Unit Circle**
Example 1: Find the exact values of the following trigonometric functions:
A. sin B. cos C. tan
D. sec E. cot F. csc
G. cot H. sec I. cos
Converting Between Radians and Degrees:
Radians to degrees, multiply by ∘ Degrees to radians, multiply by ∘
Example 2: Convert from degrees to radians or vice versa. Do not use a calculator. Simplify all solutions.
A. 240∘ B.
C. 210∘ D. 4 (radians)
When finding co-terminal angles for angles expressed in radian measure, add or subtract 2π as many
times needed. (Remember not to add 360∘, as 360∘ is not expressed in radian measure). Example 3: Sketch each angle in standard position, and find one positive and one negative co‐terminal angle for each. (Express your answers in radian measure)
A. B. 73
Example 4: Use your calculator to find the following to 3 decimal place accuracy. Check your mode. Note: When degrees are not specifically indicated, angle measures are considered to be in radians.
A. cos 3.725 B. tan47 C. csc 2.621
x
y
x
y
Study Notes 4 – Solving Trig Equations using Inverse Trigonometry (Section 8.5)
Objective: Find angles from given trigonometric ratios. Solve trigonometric equations with a calculator. Solve real‐world problems by finding angles from given trigonometric ratios. By the End of Class I Should…
So far, you have worked with trigonometric functions in a “forward” direction. You have been given angles, and you have been asked to find trigonometric ratios (in either fractional or decimal form). Suppose you were given a trig ratio and asked to find angles which produced that ratio. This is working in a “backward” direction. Many problems in the real world involve using trigonometry in a “backward” direction (ratios to angles), this is known as inverse operations. Example 1: Solve for . Example 2: A swimming pool is 20 meters long and 12 meters wide. The bottom of the pool is slanted such that the water depth is 1.3 meters at the shallow end and 4 meters at the deep end. Find the angle of depression of the bottom of the pool. The unit circle can also be used to work backwards. The question might be worded something like this: “Find all angles , in the interval 0°, 360° , that satisfy the given trigonometric equation.” Think About It: Why is a restriction given on the angle, 0°, 360° ?
When the restriction is in DEGREES your answer is in DEGREES! When the restriction is in RADIANS your answer is in RADIANS!
Example 3: Find all angles , in the interval 0°, 360° , that satisfy cos √.
Where is cosine positive?
What reference angle produces the cosine of √?
Then what angle satisfies the equation?
Inverse Operation on Your Calculator Use when finding an ANGLE
2
7
20 m 1.3 m
2.7 m
Example 4: Find all angles that satisfy the given trigonometric equation. 0°, 360° 0, 2
A. tan √3 B. sin 0
C. sin D. csc √2
E. cot 0 F. csc √
G. 2 cos 2 1 H. sin √2 sin I. sec 2 0 J. 2tan 6 0 Think About It: Find the values of for the given trigonometric equation.
cos .396 0°, 360° For trigonometric equations which are difficult or impossible to solve using algebraic methods, you can still use a calculator to solve them. Make sure to build a window which will show only the solutions that you want if solving by graphing. Example 5: Use a calculator to find the solutions to 4 cos in the interval , . Express your answers to 3 decimal place accuracy.
Study Notes 5 – Right Trig Identities (Section 8.6) Identities are true for all angles. Pythagorean Identities Reciprocal Identities Negative Identities
Ratio Identities
Co‐function Identities
Prove the following identities: 1. sin sec csc tan 2. tan cot sec csc
3. cos 4. tan
cos2 + sin2 = 1 1 + tan2 = sec2 1 + cot2 = csc2
csc1
sin
sec1
cos
cot1
tan
sin = − sin(−) csc = − csc(−)
cos = cos(−) sec = sec(−)
tan = − tan(−) cot = − cot(−)
tansincos
cotcossin
sin cos cos sin tan cot
csc sec sec csc cot tan
REMEMBER!! Don’t invent new rules. Changing things to sin and cos helps. You can’t use Pythagorean unless things are squared. Don’t move things across the equal sign when proving identities.