1. Chapter 4 Trigonometric Functions 1

2. 4.3 Right Triangle Trigonometry Objectives:  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. 2

3. 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.) 3

4. Example 1  Find the exact values of the six trig functions of the angle θ shown in the figure. 4

5. Trig Functions of Special Angles  Use special triangles to find the exact values of the trig functions of angles 45°, 30 °, and 60 °. 5 45 ° 60 ° 30 °

6. Summary of Special Angles 6  sin 30 ° = cos 60 °  sin 60 ° = cos 30 °  sin 45 ° = cos 45 ° Note:

7. 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. 7

8. Fundamental Trig Identities  List the six reciprocal identities.  List the two quotient identities.  List the three Pythagorean identities. 8

9. 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 9

10. 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 10

11. 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. 11

12. 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? 12

13. Example 5  Find the length c of the skateboard ramp shown in the figure. 13 18.4 °

14. 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. 14

15. Homework 4.3  Worksheet 4.3 15