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Pre-CalculusChapter 4

Trigonometric Functions

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4.3 Right Triangle TrigonometryObjectives: Evaluate trigonometric functions of acute angles.

Use fundamental trigonometric identities.

Use a calculator to evaluate trigonometric functions.

Use trigonometric functions to model and solve real-life problems.

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Right Triangle Definitions of Trigonometric Functions

Write the six trigonometric functions of angle θ using the right triangle shown below.

Note that θ is an acute angle. That is, 0°≤ θ ≤ 90°.(θ lies in the first quadrant.)

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Example 1 Find the exact values of the six trig

functions of the angle θ shown in the figure.

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Trig Functions of Special Angles Use special triangles to find the

exact values of the trig functions of angles 45°, 30°, and 60°.

45° 60°

30°

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Summary of Special Angles

sin 30° = cos 60°

sin 60° = cos 30° sin 45° = cos 45°

Note:

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Cofunctions Cofunctions of complementary angles are equal.

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

Cofunctions are sine & cosine, tangent & cotangent, secant & cosecant.

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Fundamental Trig Identities

List the six reciprocal identities.

List the two quotient identities.

List the three Pythagorean identities.

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Example 2 Let be θ an acute angle such that sin θ = 0.6. Use trig identities to find:

a. cos θ

b. tan θ

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Example 3 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)(sec θ – tan θ) = 1

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Angles of Elevation and Depression Angle of Elevation The angle from the horizontal upward to the object.

Angle of Depression The angle from the horizontal downward to the object.

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Example 4 A surveyor is standing 50 feet from

the base of a large tree, as shown in the figure. The surveyor measures the angle of elevation to the top of the tree as 71.5°. How tall is the tree?

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Example 5 Find the length c of the skateboard

ramp shown in the figure.

18.4°

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Example 6 In traveling across flat land you

notice a mountain directly in front of you. Its angle of elevation (to the peak) is 3.5°. After you drive 13 miles closer to the mountain, the angle of elevation is 9°. Approximate the height of the mountain.

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Homework 4.3 Worksheet 4.3