1. Copyright © 2009 Pearson Addison-Wesley 2.1-1 2 Acute Angles and Right Triangle

2. Copyright © 2009 Pearson Addison-Wesley 2.1-2 2.1 Trigonometric Functions of Acute Angles 2.2 Trigonometric Functions of Non-Acute Angles 2.3 Finding Trigonometric Function Values Using a Calculator 2.4 Solving Right Triangles 2.5Further Applications of Right Triangles 2 Acute Angles and Right Triangles

3. Copyright © 2009 Pearson Addison-Wesley1.1-3 2.1-3 Trigonometric Functions of Acute Angles 2.1 Right-Triangle-Based Definitions of the Trigonometric Functions ▪ Cofunctions ▪ Trigonometric Function Values of Special Angles

4. Copyright © 2009 Pearson Addison-Wesley1.1-4 2.2-4 Right-Triangle-Based Definitions of Trigonometric Functions For any acute angle A in standard position,

5. Copyright © 2009 Pearson Addison-Wesley1.1-5 2.2-5 Right-Triangle-Based Definitions of Trigonometric Functions For any acute angle A in standard position,

6. Copyright © 2009 Pearson Addison-Wesley1.1-6 2.2-6 Right-Triangle-Based Definitions of Trigonometric Functions For any acute angle A in standard position,

7. Copyright © 2009 Pearson Addison-Wesley1.1-7 2.2-7 Find the sine, cosine, and tangent values for angles A and B. Example 1 FINDING TRIGONOMETRIC FUNCTION VALUES OF AN ACUTE ANGLE

8. Copyright © 2009 Pearson Addison-Wesley1.1-8 2.2-8 Find the sine, cosine, and tangent values for angles A and B. Example 1 FINDING TRIGONOMETRIC FUNCTION VALUES OF AN ACUTE ANGLE (cont.)

9. Copyright © 2009 Pearson Addison-Wesley1.1-9 2.1-9 Cofunction Identities For any acute angle A in standard position, sin A = cos(90   A)csc A = sec(90   A) tan A = cot(90   A)cos A = sin(90   A) sec A = csc(90   A)cot A = tan(90   A)

10. Copyright © 2009 Pearson Addison-Wesley1.1-10 2.1-10 Write each function in terms of its cofunction. Example 2 WRITING FUNCTIONS IN TERMS OF COFUNCTIONS (a)cos 52°= sin (90° – 52°) = sin 38° (b)tan 71°= cot (90° – 71°) = cot 19° (c)sec 24°= csc (90° – 24°) = csc 66°

11. Copyright © 2009 Pearson Addison-Wesley1.1-11 2.1-11 Find one solution for the equation. Assume all angles involved are acute angles. Example 3 SOLVING EQUATIONS USING THE COFUNCTION IDENTITIES (a) Since sine and cosine are cofunctions, the equation is true if the sum of the angles is 90º. Combine terms. Subtract 6°. Divide by 4.

12. Copyright © 2009 Pearson Addison-Wesley1.1-12 2.1-12 Find one solution for the equation. Assume all angles involved are acute angles. Example 3 SOLVING EQUATIONS USING THE COFUNCTION IDENTITIES (continued) (b) Since tangent and cotangent are cofunctions, the equation is true if the sum of the angles is 90º.

13. Copyright © 2009 Pearson Addison-Wesley 2.1-13 Increasing/Decreasing Functions As A increases, y increases and x decreases. Since r is fixed, sin A increasescsc A decreases cos A decreasessec A increases tan A increasescot A decreases

14. Copyright © 2009 Pearson Addison-Wesley1.1-14 2.1-14 Example 4 COMPARING FUNCTION VALUES OF ACUTE ANGLES Determine whether each statement is true or false. (a) sin 21° > sin 18° (b) cos 49° ≤ cos 56° (a)In the interval from 0  to 90 , as the angle increases, so does the sine of the angle, which makes sin 21° > sin 18° a true statement. (b)In the interval from 0  to 90 , as the angle increases, the cosine of the angle decreases, which makes cos 49° ≤ cos 56° a false statement.

15. Copyright © 2009 Pearson Addison-Wesley 2.1-15 30°- 60°- 90° Triangles Bisect one angle of an equilateral to create two 30°-60°-90° triangles.

16. Copyright © 2009 Pearson Addison-Wesley 2.1-16 30°- 60°- 90° Triangles Use the Pythagorean theorem to solve for x.

17. Copyright © 2009 Pearson Addison-Wesley1.1-17 2.1-17 Example 5 FINDING TRIGONOMETRIC FUNCTION VALUES FOR 60° Find the six trigonometric function values for a 60° angle.

18. Copyright © 2009 Pearson Addison-Wesley1.1-18 2.1-18 Example 5 FINDING TRIGONOMETRIC FUNCTION VALUES FOR 60° (continued) Find the six trigonometric function values for a 60° angle.

19. Copyright © 2009 Pearson Addison-Wesley 2.1-19 45°- 45° Right Triangles Use the Pythagorean theorem to solve for r.

20. Copyright © 2009 Pearson Addison-Wesley 2.1-20 45°- 45° Right Triangles

21. Copyright © 2009 Pearson Addison-Wesley 2.1-21 45°- 45° Right Triangles

22. Copyright © 2009 Pearson Addison-Wesley 2.1-22 Function Values of Special Angles 60  45  30  csc  sec  cot  tan  cos  sin 