4.3 Right Triangle TrigonometryObjectives:

Evaluate trigonometric functions of acute angles

Use the fundamental trigonometric identities

Use trigonometric functions to model and solve real-life problems

Precalculus 4.3 Right Triangle Trigonometry 2

Trigonometric Functions

Given a right triangle with acute angle

sin opp

hyp

cos adj

hyp

tan opp

adj

csc hyp

opp

sec hyp

adj

cot adj

opp

opp

adj

hyp

Precalculus 4.3 Right Triangle Trigonometry 3

Example 1

· Evaluate the six trig functions of the angle shown in the right triangle

· sin =

· cos =

· tan =

· csc =

· sec =

· cot =

5

12

13

Precalculus 4.3 Right Triangle Trigonometry 4

Example 2

· Find the value of x for the right triangle shown

x

15

30°

Precalculus 4.3 Right Triangle Trigonometry 5

Example 3

· Solve ∆ABC

A

B

C

62°

6c

b

Precalculus 4.3 Right Triangle Trigonometry 6

You Try

· Solve ∆ABC

A

B

C30°

ac

15

Precalculus 4.3 Right Triangle Trigonometry 7

Special Triangles30º-60°-90°

12

30°

60°

3

sin30 sin6

cos30 cos6

tan30 tan6

sin60 sin3

cos60 cos3

tan60 tan3

Recall the conversions

30°= 60°=

6

3

1

2

3

2

1

3

3

3

3

2

1

2

3

1 3

Precalculus 4.3 Right Triangle Trigonometry 8

Special Triangles45º-45°-90°

1

145°

45°

2

sin45 sin4

tan45 tan4

Recall the conversion

45°=

4

cos45 cos4

1

2

2

2

1

2

2

2

1

11

Precalculus 4.3 Right Triangle Trigonometry 9

Co-functions of Complementary Angles

· Look back at the sines and cosines for the 30-60-90 triangle

· Note that sin 30°=cos 60°

· Co-functions of complementary angles are equivalent

sin 90 cos cos 90 sin

tan 90 cot cot 90 tan

sec 90 csc csc 90 sec

Precalculus 4.3 Right Triangle Trigonometry 10

Trigonometric Identities

· Reciprocal Identities

sin 1

csc

cos 1

sec

tan 1

cot

csc 1

sin

sec 1

cos

cot 1

tan

Precalculus 4.3 Right Triangle Trigonometry 11

Trigonometric Identities

· Quotient Identities

tan sincos

cot cossin

Precalculus 4.3 Right Triangle Trigonometry 12

Trigonometric Identities

· Pythagorean Identities

sin2 cos2 1

1 tan2 sec2

1 cot 2 csc2

Note: sin2 is (sin )2 not sin (2)

Precalculus 4.3 Right Triangle Trigonometry 13

Example 4

· Given sin=0.6, find the value of cos

· Given sin=0.6, find the value of tan

Precalculus 4.3 Right Triangle Trigonometry 14

Example 5

Use trigonometric identities to transform one side of the equation into the other.

· cossec=1

· (sec+tan)(sec-tan)=1

Precalculus 4.3 Right Triangle Trigonometry 15

Surveying

· A surveyor is standing 50 ft from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 71.5°. How tall is the tree?

Precalculus 4.3 Right Triangle Trigonometry 16

River Walk

· You are 200 yards from a river. Rather than walking directly to the river, you walk 400 yards along a straight path to the river’s edge. Find the acute angle ө between this path and the river’s edge.

Precalculus 4.3 Right Triangle Trigonometry 17

Skateboarding

· Find the length c of the skateboard ramp.

Precalculus 4.3 Right Triangle Trigonometry 18

Homework

· Star Wars Worksheet

· Page 285

44-54 even, 79-82 all