SECTION 6.3 Properties of the Trigonometric Functions 423 Finding Exact Values Using Even–Odd Properties Find the exact value of each of the following angles: (a) ( ) − ° sin 45 (b) π ( ) − cos (c) π ( ) − cot 3 2 (d) π ( ) − tan 37 4 EXAMPLE 7 Solution (a) ( ) − ° = − ° = − sin 45 sin45 2 2 ↑ Odd function (b) π π ( ) − = = − cos cos 1 ↑ Even function (c) π π ( ) − = − = cot 3 2 cot 3 2 0 ↑ Odd function (d) π π π π π ( ) ( ) − =− =− +=− =− tan 37 4 tan 37 4 tan 4 9 tan 4 1 ↑ Odd function ↑ Period is π Now Work PROBLEM 5 9 ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 6.3 Assess Your Understanding 1. The domain of the function ( ) = + + f x x x 1 2 1 is . (pp. 69–70) 2. A function for which ( ) ( ) = − f x f x for all x in the domain of f is called a(n) function. (pp. 87–89) 3. True or False The function ( ) = f x x is even. (pp. 87–89) 4. True or False The equation ( ) + = + − x x x 2 1 1 2 2 is an identity. (p. A44) Skill Building In Problems 11–26, use the fact that the trigonometric functions are periodic to find the exact value of each expression. Do not use a calculator. 11. ° sin405 12. ° cos420 13. ° tan405 14. ° sin390 15. ° csc450 16. ° sec540 17. ° cot390 18. ° sec420 19. π cos 33 4 20. π sin 9 4 21. π ( ) tan 21 22. π csc 9 2 23. π sec 17 4 24. π cot 17 4 25. π tan 19 6 26. π sec 25 6 In Problems 27–34, name the quadrant in which the angle θ lies. 27. θ θ > < sin 0, cos 0 28. θ θ < > sin 0, cos 0 29. θ θ < < sin 0, tan 0 30. θ θ > > cos 0, tan 0 31. θ θ > < cos 0, tan 0 32. θ θ < > cos 0, tan 0 33. sec 0, sin 0 θ θ < > 34. θ θ > < csc 0, cos 0 Concepts and Vocabulary 5. The sine, cosine, cosecant, and secant functions have period ; the tangent and cotangent functions have period . 6. The domain of the tangent function is . 7. Multiple Choice Which of the following is not in the range of the sine function? (a) π 4 (b) 3 2 (c) −0.37 (d) −1 8. Multiple Choice Which of the following functions is even? (a) cosine (b) sine (c) tangent (d) cosecant 9. θ θ + = sin cos 2 2 10. True or False θ θ = sec 1 sin 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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