1

Trigonometry

Basic Calculations of Angles and Sides of Right Triangles

2

Introduction

• You can use the three trig functions (sin, cos, and tan) to solve problems involving right triangles.

Using a Calculator

You can use a calculator to approximate the sine, the cosine, and the tangent of 74º. Make sure your calculator is in degree mode. The table shows some sample keystroke sequences accepted by most calculators.

Sample keystroke sequencesSample calculator

displayRounded

approximation

3.4874

0.2756

0.96130.96126169574 or 74sin Entersin

74 or 74cos cos Enter

74 or 74tan tan Enter

0.275637355

3.487414444

4

Use trigonometry to determine the length of a side of a right triangle.

5

9”

26°

x

Determining the length of a side Example 5

• In this problem, we will determine the length of side x.

6

9”

26°

xopposite

hypotenuse

adjacent

Determining the length of a side Example 5

• As always, first label the sides of the triangle...

7

9”

26°

xopposite

hypotenuse

Determining the length of a side Example 5

• Since you know the length of the hypotenuse and want to know the length of the opposite side, you should pick a trig function that contains both of them...

8

hyp

oppA sin

hyp

adjA cos

adj

oppA tan

You need to pick the sine function since it is the only one that has both the opposite side and hypotenuse in it.

Determining the length of a side Example 5

• Which trig function should you pick?

9”

26°

x

oppo

site

hypotenuse

9

9”

26°

xopposite

hypotenuse

hyp

oppA sin

926sin

xo x)438.0(9

Determining the length of a side Example 5

• Now set-up the trig function:

x"95.3 )9(9

)26(sin9xo

Use basic algebra to solve this equation. Multiply both sides of the equation by 9 to clear the fraction.

10

9”

26°

3.95”opposite

hypotenuse

Determining the length of a side Example 5

• Now you know the opposite side has a length of 3.95”.

11

75 mm

47°

x

Determining the length of a side Example 6

• Let’s try another one.

• Determine the length of side x.

12

75 mm

47°

x

hypotenuse

opposite

adjacent

Determining the length of a side Example 6

• Since the known angle (47°) will serve as your reference angle, you can label the sides of the triangle...

13

75 mm

47°

x

hypotenuse

adjacent

Determining the length of a side Example 6

• You know the length of the hypotenuse and want to know the length of the adjacent side, so pick a trig function which contains both of them...

14

hyp

oppA sin

hyp

adjA cos

adj

oppA tan

You need to pick the cosine function since it is the only one that has both the adjacent side and hypotenuse in it.

Determining the length of a side Example 6

• Which trig function should you pick?

75 mm

47°

x

hypotenuse

adja

cent

15

75 mm

47°

x

hypotenuse

adjacent

hyp

adjA cos

7547cos

xo

Use basic algebra to solve this equation. Multiply both sides of the equation by 75 to clear the fraction. )75(

75)47(cos75

xo xmm 1.51 x)682.0(75

To finish, evaluate cos 47° (which is 0.682) and multiply by 75.

Determining the length of a side Example 6

• Set-up your trig function...

16

75 mm

47°

51.1 mm

hypotenuseadjacent

Determining the length of a side Example 6

• Now you know the length of the adjacent side is 51.1 mm.

17

12 ft

35°

x

Determining the length of a side Example 7

• Let’s try a little bit more challenging problem.

• Determine the length of side x.

18

12 ft

35°

x oppositehypotenuse

adjacent

Determining the length of a side Example 7

• Label the sides of the right triangle...

19

12 ft

35°

x oppositehypotenuse

adjacent

Determining the length of a side Example 7

• Which trig function will you pick? You know the length of the side opposite and want to know the length of the hypotenuse.

20

hyp

oppA sin

hyp

adjA cos

adj

oppA tan

You need to pick the sine function since it is the only one that has both the opposite side and hypotenuse in it.

Determining the length of a side Example 7

• Which trig function should you pick?

35°

xhypotenuse

12 ft

opposite

21

12 ft

35°

x oppositehypotenuse

hyp

oppA sin

xo 12

35sin ft 9.20x

Use algebra to solve this equation. Multiply both sides of the equation by x to clear the fraction.

)(12

)35(sin xx

x o ox

35sin

12

Next, divide both sides by sin35° to isolate the unknown x.

Determining the length of a side Example 7

• Now set-up your trig function.

22

Determining the length of a side Example 8

• The reason the last problem was a little bit more difficult was the fact that you had an unknown in the denominator of the fraction.

• Keep clicking to see a similar trig function solved.

adj

oppA tan

xo 35

50tan ox

50tan

35 9.42x )(

35)50(tan x

xx o )(

35)50(tan x

xx o 35)50(tan ox

1918.1

35x

50°

35 cm

x

23

Determining the length of a side Example 9

• Your turn.

• Determine the lengths of sides x and y.

65°

45.5 mm

x

y

27

Determining the length of a side Example 9

• Since you want to know the length of side y (adjacent) and you know the length of the hypotenuse, which trig function should you select?

65°

45.5 mm

x

y

hypotenuse

opposite

adjacent

33

Determining the length of a side Example 9

• Both sides have been determined, one by trig, the other using the Pythagorean Theorem.

• Also the size of the other acute interior angle is indicated...

65°

45.5 mm

41.3 mm

19.2 mm

25°

35

Summary

• After viewing this lesson you should be able to:– Compute the length of any side of a right

triangle as long as you know the length of one side and an acute interior angle.

7.5”x60°

38

Homework pg. 469 #3,13pg. 477 # 3,6,7

pg. 469 #6,8pg. 470 #18 (no check),28 pg. 477 # 11,14

write answer only as fraction

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