60º5

?

45º8

?

Recall: How do we find “?”

65º5

?

What about this one?

60º5

?

What is the ratio of long leg to short leg?

3

1

60º11

?

60º7

?

65º5

?

65º12

?

65º123

?

These triangles are all similar (AA~).

What is the relationship of their ratios of long leg to short leg?

The ratios are all the same.

Right Triangle Trigonometry

Sections 9.1 and 9.2

What is Trigonometry?

x 70

20

x

3020

Angle ProblemTriangle Sum Theorem

Side ProblemPythagorean Theorem

x

30

20

Angle and Side Problem

Tangent Ratio

Opposite Leg of

Adjacent Leg

BCTan A

AC

AdjacentLeg

OppositeLeg

B

CAAdjacentLeg

OppositeLeg

B

CA

Opposite Leg of

Adjacent Leg

ACTan B

BC

Trig ratios are always with respect to a specific angle.

Labeling in a right triangle

a

b

B

CA

c

tan A2021

= = =oppositeadjacent

BCAC

tan B2120

= = =oppositeadjacent

ACBC

Write the tangent ratios for A and B.

Calculator Trig Functions

37°

B

CA

(37 )Tan

Make sure the calculator is set to “degrees”

Opposite

Adjacent Angle Measure

0.7536

If you must round, use at least 3 decimal places.

To measure the height of a tree, Alma walked 125 ft from the tree and measured a 32° angle from the ground to the top of the tree. Estimate the height of the tree.

The tree forms a right angle with the ground, so you can use the tangent ratio to estimate the height of the tree.

tan 32° = height125 Use the tangent ratio.

height = 125 (tan 32°) Solve for height.

125 32 78.108669 Use a calculator.

The tree is about 78 ft tall.

Sine Ratio

Opposite Leg of

Hypotenuse

BCSin A

AB

Opposite Leg of

Hypotenuse

ACSin B

AB

OppositeLeg

Hypotenuse

B

CA

OppositeLeg

Hypotenuse

B

CA

Cosine Ratio

Adjacent Leg of

Hypotenuse

ACCos A

AB

AdjacentLeg

Hypotenuse

B

CA

AdjacentLeg

Hypotenuse

B

CA

Adjacent Leg of

Hypotenuse

BCCos B

AB

Use the triangle to find sin T, cos T, sin G, and cos G. Write your answer in simplest terms.

sin T = =1220

35=

oppositehypotenuse

cos T = =1620

45=

adjacenthypotenuse

sin G = =1620

45=

opposite hypotenuse

cos G = =1220

35=

adjacent hypotenuse

Calculator Trig Functions

37°

B

CA

(37 ) 0.6018Sin

(37 ) 0.7986Cos

(37 ) 0.7536Tan

Make sure the calculator is set to “degrees”

A 20-ft. wire supporting a flagpole forms a 35˚ angle with the

flagpole. To the nearest foot, how high is the flagpole?

The flagpole, wire, and ground form a right triangle with the wire as the hypotenuse.

Because you know an angle and the measures of its adjacent side and the hypotenuse, you can use the cosine ratio to find the height of the flagpole.

cos 35° =height

20 Use the cosine ratio.

height = 20 • cos 35° Solve for height.

20 35 16.383041 Use a calculator.

The flagpole is about 16 ft tall.

SOH-CAH-TOA

Opposite Leg of

HypotenuseSin

Adjacent Leg of

HypotenuseCos

Opposite Leg of

Adjacent LegTan

SOH

CAH

TOA

SOH-CAH-TOA

Inverse Trig Functions

x°

B

CA

If the Sin of an angle is 0.8191, what is the measure of the angle?

1(0.8192)Sin

Opposite

Hypotensue Angle Measure

55

Regular vs. Inverse

(Angle measure)Opposite

TanAdjacent

1 Angle measureOpposite

TanAdjacent

(Angle measure)Opposite

SinHypotenuse

1 Angle measureOpposite

SinHypotenuse

(Angle measure)Adjacent

CosHypotenuse

1 Angle measureAdjacent

CosHypotenuse

A right triangle has a leg 1.5 units long and hypotenuse 4.0

units long. Find the measures of its acute angles to the nearest degree.

Draw a diagram using the information given.

Use the inverse of the cosine function to find m A.

cos A =1.54.0 0.375= Use the cosine ratio.

Use the inverse of the cosine.m A = cos–1(0.375)

Use a calculator.0.375 67.975687

Round to the nearest degree.m A 68

(continued)

To find m B, use the fact that the acute angles of a right triangle are complementary.

The acute angles, rounded to the nearest degree, measure 68 and 22.

m A + m B = 90 Definition of complementary angles

Substitute.68 + m B 90

m B 22

Find m R to the nearest degree.

tan R =4741 Find the tangent ratio.

So m R 49.

m R tan–1 Use the inverse of the tangent.4741

Use a calculator.48.9004944741