Grade 11 Applied Math Unit One Trigonometry Right Triangle Trigonometry

Lesson One: Introduction to Triangles Lesson Two: The Pythagorean Theorem Lesson Three: Finding a Missing Side using Trigonometric Ratios Lesson Four: Finding a Missing Angle using Trigonometric Ratios Lesson Five: Solving a Triangle Lesson Six: Applications of the Trigonometric Ratios

Oblique Triangle Trigonometry

Lesson Seven: The Law of Sines Lesson Eight: The Law of Cosines Lesson Nine: Applications of the Laws of Sines and Cosines Printed on: June 2019 Winnipeg Adult Education Centre

Trigonometry Outcomes Overview

These are the outcomes we will be learning in this unit. Check off each box once you feel confident with each outcome:

I understand how the parts of a triangle are named and labeled.

I can apply the 180o rule to determine the size of an angle in a triangle.

I can use the Pythagorean Theorem to calculate the measurement of the

third side of a right triangle.

I can figure out the length of an unknown side in a right triangle using the

sine, cosine, or tangent ratios

I can use the sine, cosine, and tangent ratios to find angle measurements in

right triangles.

I can use trigonometry to solve for all the unknown measurements in a

right triangle.

I can solve word problems that involve right triangle relationships, including

the 180o Rule, the Pythagorean Theorem and the sine, cosine and tangent

ratios.

I can solve triangle problems using the Sine Law.

I can solve triangle problems using the Cosine Law.

I can solve triangle word problems, using a variety of methods.

Using the Ti-84 Calculator

In this course we will be using the Ti-84 calculator.

Each Applied Math classroom at WAEC has a set of calculators. You will be assigned one for use in class. It is important that you reset the calculator memory at the beginning of each class. This will remove any other student’s work and give you a clean starting point. The classroom calculators are used throughout the day and must remain in the classroom. Graphing calculators can also be signed out of the library for short term use.

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Lesson One: Introduction to Triangles

A triangle is a three sided figure. Although it has three sides, it actually contains six parts: three sides and three angles. For our trigonometry unit, we will be trying to find the length of a missing side (usually called “𝑥”) or the size of a missing angle (usually called 𝜃). The symbol 𝜃 is a letter of the Greek alphabet and is pronounced “theta”. Before we start to work with the sides and the angles, there is one more fact about triangles that we need to know:

The sum of the three angles in a triangle adds up to 180°.

This means that if we know two of the angles in a triangle, we can easily find the size of the third angle (by subtracting the other two from 180°).

Examples Determine the size of θ in each of the triangles below. Round your answers to the nearest degree.

Goals:

Demonstrate an understanding of the different parts of a triangle.

Apply the 180o rule to determine the size of an angle in a triangle.

θ θ

50°

47°

62°

𝜃

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Assignment 1: Introduction to Triangles

Find the size of the unknown angle in each triangle below. Round your final answers to the nearest hundredth (2 decimal places).

35°

𝜃

1.

73⁰

𝜃

64⁰

2.

𝜃

75°

19⁰

3.

30°

𝜃 𝜃

4.

45°

𝜃

5.

20°

88°

θ

6.

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Lesson Two: The Pythagorean Theorem

A right triangle is an angle that has a measure of 90°, which means that two of its sides are perpendicular to each other. If a triangle has one right angle, it is called a right triangle. A small box is placed where the two perpendicular sides meet. This box means that the angle created by the two perpendicular sides is 90°. A 90° is also referred to as a right angle. Triangles with a 90°angle are called ‘right angle triangles’.

Why is it impossible to have more than one right angle in a triangle? In a right triangle, special vocabulary is used. The longest side, found across from the 90° angle, is called the hypotenuse. The two shorter sides that meet at the 90o angle are called the legs or sides. Label the hypotenuse on the triangles below.

Goals:

Calculate the measurement of the third side of a right triangle, using the Pythagorean Theorem.

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Pythagoras was a Greek philosopher and mathematician who is credited with discovering that the squares of the sides of a right triangle add up to the square of the hypotenuse. We can use this knowledge to find the length of a missing side in a right triangle. The Pythagorean Theorem is often written as 𝑎2 + 𝑏2 = 𝑐2 where 𝑐 is the hypotenuse and 𝑎 and 𝑏 are the sides. We can also write the Pythagorean Theorem as

𝑠𝑖𝑑𝑒2 + 𝑠𝑖𝑑𝑒2 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒2.

As a result of the Pythagorean Theorem, when we know the lengths of two sides, we can find the length of the missing side.

Example 1 Use the Pythagorean Theorem to find the length of the missing side. (The missing

side is labelled with an 𝒙).

Example 2 Find the length of the missing side (side 𝒙).

𝑥

8.7 cm

6.5 cm

𝑥

15.67 km

7.6 km

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Assignment 2: The Pythagorean Theorem

Use the Pythagorean Theorem to find the length of side 𝑥 in each of the triangles below. Round your final answers to the nearest hundredth (2 decimal places).

21 cm

15 cm 𝑥

1.

2.9 m

𝑥 4.3 m

4.

19 cm

8 cm

𝑥

2.

40 cm

30 cm 𝑥

3.

21 m

𝑥

17 m

7.

92 cm 𝑥

68 cm

5.

3.8 km

6 km

𝑥 6.

16 cm

21 cm 𝑥

8.

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Lesson Three: Using Trigonometric Ratios to Find Side Lengths

When we know two of the three side lengths in a right triangle, we can find the third side length using the Pythagorean Theorem (as we did in the last lesson). But, if we only know the length of one side, we can’t use the Pythagorean Theorem. There are 3 more formulas we can use to find missing side lengths. These are called the trigonometric ratios:

sine 𝜃 =𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 cosine 𝜃 =

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 tangent 𝜃 =

𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

Before we can use the above formulas, we first have to understand the concepts of “opposite” and “adjacent”. In the triangles below, notice that the side across from the 𝜃 is labelled the opposite side, the side close to 𝜃 is labelled the adjacent side. The hypotenuse, as always, is the side across from the 90° angle.

Goals:

Determine the length of an unknown side length in a right triangle using the sine, cosine, or tangent ratios.

𝜃

17 m ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

8 m 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒

15 m 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒

𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

𝜃

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Label the sides of the triangles shown below as opposite, adjacent, and hypotenuse.

Now that we are familiar with the vocabulary we return to our formulas:

sine 𝜃 =𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 cosine 𝜃 =

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 tangent 𝜃 =

𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

These three formulas are used when solving right triangle problems. Note that each formula has a different combination of sides forming the ratio:

𝑠𝑖𝑛 𝜃 =𝑜𝑝𝑝

ℎ𝑦𝑝 The sine of an angle is equal to the length of the opposite side over the length of

the hypotenuse.

𝑐𝑜𝑠 𝜃 =𝑎𝑑𝑗

ℎ𝑦𝑝 The cosine of an angle is the length of the adjacent side over the length of the

hypotenuse.

𝑡𝑎𝑛 𝜃 =𝑜𝑝𝑝

𝑎𝑑𝑗 The tangent of an angle is the length of the opposite side over the length of the

adjacent side.

Many people use “SOH CAH TOA” to remember the correct combination (and order) of sides for each formula.

𝜃 𝜃

𝜃 𝜃

𝜃

𝜃

𝜃

𝜃 𝜃

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The following examples show how to use the trigonometric ratios to determine a side length in a right triangle.

Example 1

Example 2

29 mm

72°

𝑥

27 feet

55°

𝑥

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Assignment 3: Using Trigonometric Ratios to Find Side Lengths

Determine the length of side 𝑥 using one of the trigonometric ratios. Where needed, round final answers to the nearest hundredth of a unit.

38°

𝑥

5 m

1.

64°

11 km 𝑥

2.

19 feet

43°

𝑥 3.

𝑥

73°

56 mm

4.

27 in

𝑥

37° 5.

𝑥

18 yards

24°

6.

59°

15 m

𝑥 7.

𝑥

29 cm

20°

8.

27°

𝑥 54 miles

9.

41°

38 km

𝑥 10.

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Lesson Four: Using Trigonometric Ratios to Find

Angle Measures

sin 𝜃 =𝑜

ℎ cos 𝜃 =

𝑎

ℎ tan 𝜃 =

𝑜

𝑎

Trig ratios can be used to find a missing angle in a right triangle. The trig ratio is chosen and set up in the same manner, but the calculator function is different. Your teacher will help you through the following examples.

Example 1: Solve for 𝜃.

Example 2: Solve for 𝜃.

Example 3: Solve for 𝜃.

Goals:

Determine the size of an unknown angle in a right triangle using the sine, cosine, or tangent ratios.

41 m 𝜃

26 m

19.7 km

𝜃

42.5 km

21’

13’

𝜃

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Assignment 4: Using Trigonometric Ratios to Find Angle Measures

Determine the size of angle 𝜃 using one of the trigonometric ratios. Round your answers to the nearest hundredth of a degree.

5 ft

𝜃

4 ft

1.

11 mm 𝜃

9 mm

2.

18 cm

𝜃

19 cm

3.

47 mi

27 mi

𝜃 5.

56 yd

𝜃

64 yd

4.

33 km

𝜃

18 km

6.

15 cm

𝜃

13 cm 7.

23 ft

𝜃

29 ft

8.

54’

𝜃

42’

9.

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Lesson Five: Solving a Triangle

To solve a triangle means that you must find the lengths of all the unknown sides and the sizes of all of the unknown angles. Usually, we are given three measurements and have to solve for the remaining three parts of the triangle.

Example 1 Solve triangle LMN.

Example 2 Solve triangle PQR.

28 km 32 km

P

Q

R

L

N

13.5 cm

42o M

Goals:

Use trigonometry to solve for all the unknown side lengths and angle measures in a right triangle.

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Assignment 5: Solving a Triangle

Solve the following triangles. Where necessary, round side lengths to the two decimal places and round angle measurements to the nearest degree.

37’

42’

A

C B

1.

26 mm

59° E F

D

3.

Q

62°

24 “

P

R

4.

K

51 in

42 in G

H

2.

N

29 cm

14 cm

L M

5.

T

19°

151 yds

U

S

6.

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Lesson Six: Applications of the Trigonometric Ratios

Example 1 Kyle stands 8 feet from the base of a street light. The angle of elevation from the

ground to the top of the street light is 75o. What is the height of the street light?

Example 2 A 6.12 m long conveyor belt reaches a height of 1.2 metres at its higher end.

Determine the angle of inclination.

Example 3. The angle of depression from the top of a viewing tower to an animal on the

ground is 30o. The height of the tower is 25 metres. How far is the animal from the top of the viewing tower?

Goals:

Solve word problems that involve right triangle relationships, including: o the 180° rule. o the Pythagorean Theorem. o the 3 trigonometric ratios.

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Assignment 6: Applications of the Trigonometric Ratios

Solve each of the following problems. When rounding is needed, express side lengths or distances to the nearest hundredth and express angles to the nearest degree.

1. When the angle of elevation of the sun is 68°, a tree casts a shadow on the ground that is

14.3 metres long. How tall is the tree? 2. A kite string is 350 m long. The angle the string makes with the ground is 50°. How far from

the person holding the string is a person standing directly under the kite? 3. A 9 foot ladder leans against the side of a wall. The bottom of the ladder is 1.5 feet from the

base of the wall. Find the measure of the angle between the ladder and the ground. 4. A theme park has a very long slide. The slide covers a horizontal distance of 1200 m and a

vertical distance o f 213 m. a) How many metres long is the slide?

b) Determine the angle of inclination of the slide. 5. The shadow of the Calgary Tower measures 84 m when the angle of elevation of the sun is

66°. Find the height of the Calgary Tower. 6. A ship is on the surface of the water, and its radar detects a submarine at a distance of 238

feet, at an angle of depression of 23°. How deep underwater is the submarine? 7. An observer is 120 feet from the base of a TV tower, which is 150 feet tall. Find the angle of

elevation to the top of the tower from the point where the observer is standing.

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Lesson Seven: The Law of Sines

Triangles that do not have a right angle are called oblique triangles. For oblique triangles, we can’t use the same trigonometric ratios that we used for right triangles. For oblique triangles, we use the Law of Sines and the Law of Cosines to solve for unknown side lengths and angle measures. We also label oblique triangles differently. We label the angles (at the vertices or corners) with capital letters and the sides opposite each angle are labelled as their ‘partner’ with the same letter in lower case. Notice in the triangle below that Angle 𝐴 ‘partners’ with side 𝑎; Angle 𝐵 ‘partners’ with side 𝑏, and Angle ‘partners’ with side 𝑐.

There is a relationship between the angles and the sides opposite them. This relationship is called the Sine Law.

Although there are 3 ratios for each triangle, we only use 2 of them at a time. We can use a ‘cross multiply’ method with any two of the ratios of the Law of Sines to find an unknown side length or angle measure in an oblique triangle.

𝑐

B 𝐶

𝐴

𝑎

𝑏

Law of Sines

𝑎

sin 𝐴=

𝑏

sin 𝐵=

𝑐

sin 𝐶

Goals:

Use the Law of Sines to solve for unknown sides and angles of oblique triangles.

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Using the Law of Sines to find an Unknown Side Length

Example 1: Finding the length of an unknown side using the Law of Sines Find the length of side 𝑥 in this triangle.

Example 2 Solve for 𝑥.

Example 3 Solve for 𝑥.

26 ft

62° 43°

𝑥

46 m

35° 76°

𝑥

52 cm

80°

29°

𝑥

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Using the Law of Sines to find an Unknown Angle Measure The Law of Sines can also be used to find the size of an angle. The set-up is the same but the use of the calculator is a bit different. Your teacher will explain this while doing these examples.

Example 4:

Determine the size of angle 𝜽 in the following triangle.

Example 5 Solve for 𝜃.

Example 6 Solve for 𝜃.

27 km

60°

𝜃

36 km

𝜃

39 m 62 m

33°

𝜃

51 m 79 m

28°

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Assignment 7: Using the Law of Sines to find an Unknown Side Length or Angle Measure

Use the Sine Law to solve for the length of side 𝑥 or measure of angle 𝜃 in the following triangles. Express your final answers in the nearest hundredth.

53°

12 m 𝑥

39°

𝜃

100 km 73 km

50°

𝜃

43 ft

15 ft 37°

25’

𝑥

70°

40°

52 cm 𝑥

51° 48°

35 km 23 km

40° 𝜃

1. 2.

3. 4.

5. 6. 7.

22°

𝑥

58°

173 in

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Lesson Eight: The Law of Cosines

We can only use the Law of Sines when we have a ‘complete partnership’. This means that we know both the length of a side and the measure of its partner angle. There are certain triangle problems where we do not know an angle and its opposite side. For these problems, we use the Law of Cosines. Although this law seems more complicated at first, after we have used it a few times, it becomes much easier. We label the triangle the same way we labelled it when we used the Law of Sines.

There are two versions of the Law of Cosines. One version is used to solve for an unknown side length. The other version is used to solve for an unknown angle measure.

The Law of Cosines

To solve for a side, use:

𝒂𝟐 = 𝒃𝟐 + 𝒄𝟐 − 𝟐𝒃𝒄𝑪𝒐𝒔𝑨

To solve for an angle, use:

𝐶𝑜𝑠 𝐴 =𝑏2 + 𝑐2 − 𝑎2

2𝑏𝑐

Goals:

Use the Law of Cosines to solve for unknown sides and angles of oblique triangles.

𝑐

B 𝐶

𝐴

𝑎

𝑏

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Using the Law of Cosines to find the Length of an Unknown Side

Example 1: Find the length of side 𝑥.

Example 2 Find the length of side 𝑥.

7 m

5 m

51°

𝑥

23 feet 15 feet

29°

𝑥

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Using the Law of Cosines to find the Measure of an Unknown Angle Just as with the Law of Sines, we can also use the Law of Cosines to find the size of an angle. In such cases, we must know all three sides and no angles.

Example 3 Determine the size of θ, to the nearest degree.

Example 4

9 km

7 km 6 km

𝜃

124 km

103 km 87 km

𝜃

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Assignment 8: The Law of Cosines Assignment

Using the triangles below, solve for side x or angle θ to the nearest hundredth.

1. 30 cm

51 cm

85°

𝑥

7 yds

6 yds

53°

𝑥

2.

12 ft

17 ft 20 ft

𝜃

3.

19 m

22 m 28 m

𝜃

4.

𝑥

74’

30° 5. 13 yds

26 yds 23 yds

𝜃

6.

10 km

12 km 15 km

𝜃

7.

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Assignment 9: Applications of the Trigonometry

For each of the following questions, draw and label a diagram and then solve the problem. Round sides to the nearest hundredth and angles to the nearest whole number. 1. A triangular garden had sides measuring 15 m, 18 m, and 20 m in length. Find the size of the

smallest angle. 2. Two lifeguards saw a swimmer in danger. The swimmer is between the two lifeguards. The

lifeguards were 50 m apart. The angle from one lifeguard to the swimmer was 83° and the angle from the other lifeguard to the swimmer was 61°. How far was the closest lifeguard to the swimmer?

3. Two boats leave a dock and the angle between their paths is 63°. An hour later, one boat has traveled 3.8 km while the other has traveled 5.2 km. How far apart are the boats at this time?

4. David runs 4 km in a straight line from home. He turns through an angle of 115° and runs

an additional 3 km. Find the direct distance from his current position to his home. 5. An observer is standing 100 ft away from a building. The angle of elevation from the observer

to the top of the building is 41°. There is a poster on the side of the building. The angle of elevation from the observer to the poster is 21°. How far is the poster from the top of the building?

6. A roof is shown as in the diagram below:

a) Find the width of the base of the roof. b) Find the height of the roof.

23 m

23 m 130°

4 km Home

3 km

115°

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Answers to Assignments Assignment 1: Introduction to Triangles 1. 55° 2. 43° 3. 86° 4. 75° 5. 45° 6. 72°

Assignment 2: Pythagorean Theorem 1. 25.81 cm 2. 20.62 cm 3. 50 cm 4. 3.17 m 5. 61.97 cm 6. 4.64 km 7. 27.02 m 8. 26.40 cm

Assignment 3: Finding a Missing Side Length 1. 3.94 m 2. 5.37 km 3. 13.90 ft 4. 183.17 mm 5. 44.86 in 6. 44.25 yards 7. 12.86 m 8. 10.56 cm 9. 48.11 miles 10. 57.92 km

Assignment 4: Finding a Missing Angle 1. 37° 2. 51° 3. 19° 4. 49° 5. 35° 6. 33° 7. 30° 8. 38° 9. 39° Assignment 5: Solving a Triangle 1. c = 55.97’, A = 48.62°, B = 41.38° 2. g = 28.93 in, G = 34.56°, H = 55.44° 3. f= 30.33 mm, d = 15.62 mm, D = 31° 4. q = 12.76”, r = 27.18”, Q = 28° 5. m = 25.40 cm, M = 61.13°, L = 28.87° 6. t = 51.99 yards, u = 159.70 yards, S = 71°

Assignment 6: Right Angle Triangle Applications 1. 35.39 m 2. 224.98 m 3. 80.41° 4.a) 1218.76 m b) 10.07° 5. 188.67 m 6. 92.99 feet 7. 51.34°

Assignment 7: The Law of Sines 1. 15.23 m 2. 17.10’ 3. 49.72 cm 4. 78° 5. 76.42 in 6. 34° 7. 12°

Assignment 8: The Law of Cosines 1. 56.87 cm 2. 5.87 yards 3. 37° 4. 43° 5. 38.31’ 6. 62° 7. 53°

Assignment 9: Applications of the Laws of Sines and Cosines 1. 46.13° 2. 74.40 m 3. 4.85 km 4. 5.93 km 5. 48.54 ft 6.a) 41.69 m b) 9.72 m

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Trigonometry Outcomes Summary These are the outcomes that have been covered in this unit. Check off each box once you feel confident that you can demonstrate that skill:

I understand how the parts of a triangle are named and labeled.

I can apply the 180o rule to determine the size of an angle in a triangle.

I can use the Pythagorean Theorem to calculate the measurement of the

third side of a right triangle.

I can figure out the length of an unknown side in a right triangle using the

sine, cosine, or tangent ratios

I can use the sine, cosine, and tangent ratios to find angle measurements in

right triangles.

I can use trigonometry to solve for all the unknown measurements in a

right triangle.

I can solve word problems that involve right triangle relationships, including

the 180o Rule, the Pythagorean Theorem and the sine, cosine and tangent

ratios.

I can solve triangle problems using the Sine Law.

I can solve triangle problems using the Cosine Law.

I can solve triangle word problems, using a variety of methods.

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Ongoing Self-Assessment for Mathematics Students

Understanding How confident are you in your ability to demonstrate understanding of the outcomes of this unit? My ability to demonstrate understanding is a: STRENGTH CHALLENGE

Attendance Did you have consistently good attendance during this unit?

My attendance is a: STRENGTH CHALLENGE

Out of Class Practice Did you feel that when you needed to practice a math skill outside of class, you were able to do so?

My ability to practice outside of class time is a: STRENGTH CHALLENGE

Accessing Help If you answered CHALLENGE to any of the questions above, consider the following options for accessing help in order to be more successful in this course:

Talk to your TEACHER.

Make time to visit the RESOURCE ROOM (ROOM 104).

Get help / support / materials from a CLASSMATE.

Use any resources provided on a CLASS BLOG (if available). You have completed a unit in this Math course. Please take some time to reflect on your thoughts regarding your academic strengths and challenges as they relate to the outcomes of this unit. You can also reflect on any previous outcomes of this course. __________________________________________________________________ __________________________________________________________________