Trigonometry 10.1

• Define trigonometry.

• Find the ratio of the sides in a right triangle.

• Use trigonometry to find the measures of unknown sides and angles in right triangles.

• Use a graphing calculator to find the measures of unknown sides and angles.

• Label the sides and angles of a right triangle correctly.

• Solve a right triangle.

Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement.

Definition

Triangle Labeling

All angles of a triangle are uppercase letters and the sides opposite them are the corresponding lower case letters.

Calculator for Homework

Make sure DEG is shown in the top left corner.

To evaluate trig functions of acute angles other than 30, 45, and 60, you will use the calculator.

Your calculator has keys marked Sin, Cos, and Tan.

**Make sure the MODE is set to the correct unit of angle measure. (Degree vs. Radian)

Using the calculator to evaluate trig functions

Graphing Calculator

Find each value using a calculator. Round to the nearest ten-thousandths degrees.

a. Sin 43°b. Cos 84°c. Tan 15°

.6820.1045.2679

Calculator Work

d. Sin 36°e. Cos 50°f. Tan 38°

.5878.6428.7813

g. Sin 17°h. Cos 75°i. Tan 26°

.2924.2588.4877

j. Sin 56°k. Cos 22°l. Tan 43°

.8290.9272.9325

Using Inverse Trigonometric Functions to Find AnglesUse a calculator to find an angle A in degrees that satisfies

sin A .9677091705.

Calculator Work

SolutionWith the calculator in degree mode, we find that an angle having a sine value of .9677091705 is 75.4º. Write this as sin-1 .9677091705 75.4º.

Find each value using a calculator. Round to the degree.

a. Sin A = .8829

b. Cos A = .5

c. Tan A = .4663

62

60

25

Calculator Work

d. Sin B = .2588

e. Cos B =.5592

f. Tan B = 2.0503

15

56

64

Trig Definitions

• Sin (angle) =

• Cos (angle) =

• Tan (angle) =

Opposite----------------Hypotenuse

Adjacent----------------Hypotenuse

Opposite ---------------- Adjacent

S-O-H

C-A-H

T-O-A

hypotenuse

In a right triangle, the shorter sides are called legs and the longest side (which is the one opposite the right angle) is called the hypotenuse

a

b

c

We’ll label them a, b, and c and the angles A,B and C. Trigonometric functions are defined by taking the ratios of sides of a right triangle.

B

A

First let’s look at the three basic functions.

SINECOSINE

TANGENT

They are abbreviated using their first 3 letters

c

a

hyp.

opp.Asin

opposite

c

b

hyp.

adj.Acos

b

a

adj.

opp.Atan

C

adjacent

oppositehypotenuse

SinOpp

Hyp

Leg

adjacent

CosAdj

Hyp

Leg

TanOpp

Adj

Leg

Leg

hypotenuseopposite

adjacent

Sin, Cos, or Tan?

x

7

35o

S H

OC H

A

T A

O

Answer: Tan

You know the adjacent and want the opposite.

10x

40o

S H

OC H

A

T A

O

Answer: Sin

You know the opposite and want the hypotenuse.

Sin, Cos, or Tan?

20

35o

S H

OC H

A

T A

O

Answer: Cos

You know the adjacent and want the hypotenuse.

x

Sin, Cos, or Tan?

x12

38o

S H

OC H

A

T A

O

Answer: Sin

You know the hypotenuse and want the opposite.

Sin, Cos, or Tan?

x

21o

S H

OC H

A

T A

O

Answer: Cos

You know the hypotenuse and want the adjacent.

100

Sin, Cos, or Tan?

1018

o

S H

OC H

A

T A

O

Answer: Sin

You know the opposite and the hypotenuse. You want to find the angle.

Sin, Cos, or Tan?

24

x

37o

S H

OC H

A

T A

O

Answer: Tan

You know the opposite and want the adjacent.

Sin, Cos, or Tan?

10

15

o

S H

OC H

A

T A

O

Answer: Sin

You know the opposite and the hypotenuse. And want to know the angle

Sin, Cos, or Tan?

20

x

42o

S H

OC H

A

T A

O

Answer: Tan

You know the opposite and want the adjacent.

Sin, Cos, or Tan?

200400

o

S H

OC H

A

T A

O

Answer: Sin

You know the opposite and the hypotenuse. You want to find the angle.

Sin, Cos, or Tan?

Find the values of sin A, cos A, and tan A; sin B, cos B, and tan B in the right triangle.

Solution

SOH CAH TOA

6

810

SOH CAH TOA

10

8

10

6

6

8

Hyp

AdjACos

Adj

OppATan

4

5

3

5

4

3

Hyp

OppASin

A

B

C

70

24

θ

SOH CAH TOA

Find c. a2 + b2 = c2 242 + 702 = c2 5476 = c2 c = 74

74

Find the values of the trigonometric functions for θ.

SOH CAH TOA

Find a. a2 + 102 = 262 a2 + 100 = 676 576 = c2 c = 24

24

– Step 1: Draw a triangle to fit problem

– Step 2: Label sides from angle’s view

– Step 3: Identify trig function to use

– Step 4: Set up equation

– Step 5: Solve for variable

To Solve Any Trig Word Problem

Assignment

8.3 Practice 1 – 158.3 Practice 1 – 15

Solve the triangle.

55 °

16 fty

Solve means to find all angles and all sides.x

a. Sin 55 = b. Cos 55 =

c. mB =

y 13.11 ft x 9.18 ft

35

A

B

C

16

y

16

x

From a point 80m from the base of a tower, the angle from the ground is 28˚. How tall is the tower?

80

28˚

x

Using the 28˚ angle as a reference, we use opposite and adjacent sides.

Use tan 28˚ = x

80oppadj

80 (tan 28˚) = x

80 (.5317) = x x ≈ 42.5 m

tan

A ladder that is 20 ft is leaning against the side of a building. If the angle formed between the ladder and ground is 75˚, how far is the bottom of the ladder from the base of the building?

ladd

er

bu

ildin

g

20

x75˚

Using the 75˚ angle as a reference, we use hypotenuse and adjacent side.

Use cos 75˚ = x

20adjhyp

20 (cos 75˚) = x

20 (.2588) = x x ≈ 5.2 ft

cos

Find the missing value.Find the measure of the missing side or hypotenuse for the triangle.

L

M

N

8

35°

x

a.a. b.b. c.c.

d.d. e.e. f.f.41.0441.04

13.9513.95

14.1414.14 42.4342.43

3737 184.08184.08

Find the missing valueFind the measure of the missing side or hypotenuse for the triangle.

a.a. b.b.

c.c.

6651.87 ft6651.87 ft 15.45 ft15.45 ft

137.97 ft137.97 ft d.d. 16.48 ft16.48 ft

x

Find the missing valueFind the measure of the missing side or hypotenuse for the triangle.

a.a. b.b.106.48 ft106.48 ft 135.32 ft135.32 ft

c.c. 4.95 ft4.95 ft d.d. 8398.54 ft8398.54 ft

Find the missing valueFind the measure of the missing side or hypotenuse for the triangle.

a.a. b.b. c.c.72.79 m72.79 m 74.89 ft74.89 ft 445.38 ft445.38 ft

d.d. e.e. f.f.524.46 m524.46 m 355.77 m355.77 m 3090.96 ft3090.96 ft

Find the missing value.Find the measure of the missing angle.

6 ft Angle A

a.a. b.b.

c.c.

6060 4.764.76

15.9515.95

Things to remember.

To solve a triangle find all missing sides an angles.

Use inverse trigonometric functions to find a missing angle.

AssignmentGeometry:

8.3 Practice 16 – 23

Back 13, 14

Angles of Elevation & Depression 10.2

DefinitionsAngle of elevation is the angle between the horizontal and the line of sight to an object above the horizontal.

Angle of depression is the angle between the horizontal and the line of sight to an object below the horizontal.

Angles of Elevation and Depression

Since the two horizontal lines are parallel, by Alternate Interior Angles the angle of depression must be equal to the angle of elevation.

Bottom Horizontal

Top Horizontal

Line of Sight

Angle of Elevation

Angle of Depression

Classify the angles as an angle of elevation or an angle of depression.

1

1 is formed by a horizontal line and a line of sight to a point below the line. It is an angle of depression.

4

4 is formed by a horizontal line and a line of sight to a point above the line. It is an angle of elevation.

Use the diagram to classify the angles as an angle of elevation or depression.

5

6

6 is formed by a horizontal line and a line of sight to a point above the line. It is an angle of elevation.

5 is formed by a horizontal line and a line of sight to a point below the line. It is an angle of depression.

Classify the angles as an angle of elevation or depression.

6

9

angle of depression

angle of elevation

When the sun is 62˚ above the horizon, a building casts a shadow 18 m long. How tall is the building?

x

18shadow

62˚

Using the 62˚ angle as a reference, we use opposite and adjacent side.

Use tan 62˚ = x18

oppadj

18 (tan 62˚) = x

18 (1.8807) = x x ≈ 33.9 m

tan

A kite is flying at an angle of elevation of about 55˚. Find the height of the kite if 85m of string has been let out.

string

85x

55˚

kite

Using the 55˚ angle as a reference, we use hypotenuse and opposite side.

Use sin 55˚ = x

85opphyp

85 (sin 55˚) = x

85 (.8192) = x x ≈ 69.6 m

sin

A 5.50 foot person standing 10 feet from a street light casts a 14 foot shadow. What is the height of the streetlight?

5.5

14 shadowx˚

tan x˚ = 5.5

14

x° ≈ 21.45° About 9.4 ft.

tan 21.4524

height

10

The angle of depression from the top of a tower to a boulder on the ground is 38º. If the tower is 25m high, how far from the base of the tower is the boulder?

25

x

angle of depression38º

38º

Using the 38˚ angle as a reference, we use opposite and adjacent side.

Use tan 38˚ = 25/x oppadj

(.7813) = 25/x

x = 25/.7813 x ≈ 32.0

tan

Alternate Interior Angles are congruent

32°

P

A 2 mi

x

J

Jody sees a plane above the airport at an angle of elevation of 32°. She is 2 miles from the airport where it is circling. How high is the airplane above the airport?

adj

oppB tan

232tan

x x32tan2 x25.1

The plane is approximately 1.25 miles above the airport.

T1

T2

8,000 ft1,000 ft.

X°

A forestry service has two fire towers located 8,000 feet apart. If the first is located 1,000 feet above on a mountain, what is the angle of depression from the first to the second tower?

hyp

oppX sin

8000

1000sin X

125.sin X

)125(.sin 1X

18.7X

The angle of depression from the first tower to the second is about 7.18°.

Suppose the ranger sees another fire and the angle of depression to the fire is 3°. What is the horizontal distance to this fire? Round to the nearest foot.

Write a tangent ratio.

Multiply both sides by x and divide by tan 3°.

x 1717 ft Simplify the expression.

By the Alternate Interior Angles Theorem, mF = 3°.

3°

3

x

Solve for x.

a. Tan 12 =

b. Tan 3 =

y .6377 km

x .3144 mi

3

x

6

x

AssignmentGeometry:

8.4 Angle Elevation & Depression

1 – 13