4.3 Right Triangle Trigonometry

Objectives:• Evaluate trigonometric functions of acute angles

• Use trig identities• Evaluate trig functions with a calculator

• Use trig functions to model and solve real life problems

Right Triangle Trigonometry

hypotenuse

θ

Side adjacent to θ

Side opposite θ

Using the lengths of these 3 sides, we form six ratios that define the six trigonometric functions of the acute angle θ.

sine cosecantcosine secanttangent cotangent

*notice each pair has a “co”

Trigonometric Functions

• Let θ be an acute angle of a right triangle.

hypopp

sinhypadj

cosadjopp

tan

opphyp

cscadjhyp

secoppadj

cot

RECIPROCALS

Evaluating Trig Functions

• Use the triangle to find the exact values of the six trig functions of θ.

hypotenuse

θ

3

4

Special Right Triangles

45-45-90 30-60-90

45°

45°

1

1

2

30°

60°

21

3

Evaluating Trig Functions for 45°

• Find the exact value of sin 45°, cos 45°, and tan 45°

Evaluating Trig Functions for 30° and 60°

• Find the exact values of sin60°, cos 60°, sin 30°, cos 30°

30°

60°

Sine, Cosine, and Tangent of Special Angles

21

6sin30sin 0

22

4sin45sin 0

23

3sin60sin 0

23

6cos30cos 0

22

4cos45cos 0

21

3cos60cos 0

31

6tan30tan 0

14

tan45tan 0

33

tan60tan 0

sin30° = ½ = cos60° (notice that 30° and 60° are complementary angles)

sin(90° - θ) = cos θ cos(90° - θ) = sin θ

tan(90° - θ) = cot θ cot(90° - θ) = tan θ

sec(90° - θ) = csc θ csc(90° - θ) = sec θ

Trig Identities

• Reciprocal Identities

csc1sin

sec1cos

cot1tan

sin1csc

cos1sec

tan1cot

Trig Identities (cont)

• Quotient Identities

• Pythagorean Identities

cossintan

sincoscot

1cossin 22

22 sectan1

22 csccot1

Applying Trig Identities

• Let θ be an acute angle such that sin θ = .6. Find the values of (a) cos θ and (b) tan θ using trig identities.

Using Trig Identities

• Use trig identities to transform one side of the equation into the other (0 < θ < π/2)

a) cos θ sec θ = 1

b) (sec θ + tan θ)(secθ – tanθ) = 1

Evaluating Using the Calculator

• sin 63°

• tan (36°)

• sec (5°)

Applications of Right Triangle Trigonometry

• Angle of elevation: the angle from the horizontal upward to the object

• Angle of depression: the angle from the horizontal downward to the object

Word Problems

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

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

• Find the length c of the skateboard ramp.