14 Chapter Test

Do you know HOW?

Simplify each trigonometric expression.

1. sin θ + cos θ cot θ sine theta plus cosine theta co-tangent theta
2. sec θ sin θ cot θ secant theta sine theta co-tangent theta
3. cot θ ( tan θ + cot θ ) co-tangent theta open tangent theta plus co-tangent theta close

Verify each identity.

4. sec θ sin θ cot θ = 1 secant theta sine theta co-tangent theta equals 1
5. csc 2 θ − cot 2 θ = 1 co-secant squared , theta negative , co-tangent squared , theta equals 1
6. sec θ cot θ = csc θ secant theta co-tangent theta equals co-secant theta
7. sec 2 θ − 1 = tan 2 θ secant squared , theta negative 1 equals , tangent squared , theta

Use a unit circle and 30°-60°-90° triangles to find values of θ theta in degrees for each expression.

8. sin θ = 3 2 sine theta equals , fraction square root of 3 , over 2 end fraction
9. cos θ = 3 2 cosine theta equals , fraction square root of 3 , over 2 end fraction
10. cos θ = − 1 cosine theta equals negative 1
11. tan θ = 3 tangent theta equals square root of 3

Solve each equation for θ theta with 0 ≤ θ < less than or equal to , theta less than 2π.

12. 4 sin θ + 2 3 = 0 4 sine theta plus 2 square root of 3 equals 0
13. 2 cos θ = 1 2 cosine theta equals 1
14. 2 sin θ − 1 = 0 square root of 2 sine theta negative 1 equals 0

In Δ ABC , cap delta find each value as a fraction and as a decimal. Round to the nearest hundredth.

15. sin A
16. sec A
17. cot A
18. csc C
19. sec C
20. tan C

In Δ DEF , ∠ F cap delta is a right angle. Find the remaining sides and angles. Round your answers to the nearest tenth.

21. e = 6 , f = 10 e equals 6 comma f equals 10
22. d = 10 , e = 12 d equals 10 comma e equals 12
23. e = 21 , f = 51 e equals 21 comma f equals 51
24. d = 5.5 , e = 2.6 d equals 5.5 comma e equals 2.6

25. Find the area of the triangle.

    

26. In Δ ABC , m ∠ A = 45 ° , m ∠ C = 23 ° , cap delta and B C = 25 b c equals 25 in. Find AB to the nearest tenth.
27. In Δ MNP , m ∠ N = 45 ° , m = 20 cap delta cm, and p = 41 p equals 41 cm. Find n to the nearest tenth.
28. In Δ PQR , p = 51  ft  , q = 81 cap delta ft, and r = 61 ft. r equals 61 , ft. Find m ∠ R m angle r to the nearest tenth.
29. In Δ STU , m ∠ S = 96 ° , t = 8 cap delta in., and u = 10 u equals 10 in. Find m ∠ U m angle u to the nearest tenth.

Verify each identity.

30. − sin ( θ − π 2 ) = cos θ negative sine . open . theta minus , pi over 2 . close . equals cosine theta
31. csc ( θ − π 2 ) = − sec θ co-secant . open . theta minus , pi over 2 . close . equals negative secant theta

Solve each trigonometric equation for θ theta with 0 ≤ θ < 2 π . 0 less than or equal to theta less than 2 pi .

32. sin ( θ − π 2 ) = sec θ sine . open . theta minus , pi over 2 . close . equals secant theta
33. cot ( π 2 − θ ) = sin θ co-tangent . open . pi over 2 , minus theta . close . equals sine theta

Use a double-angle identity to find the exact value of each expression.

34. sin 60 ° sine , 60 degrees
35. cos 60 ° cosine , 60 degrees
36. tan 60 ° tangent , 60 degrees

Use a half-angle identity to find the exact value of each expression.

37. tan 30 ° tangent , 30 degrees
38. sin 90 ° sine , 90 degrees
39. cos 180 ° cosine , 180 degrees

Do you UNDERSTAND?

40. Writing Suppose you know the lengths of all three sides of a triangle. Can you use the Law of Sines to find the measures of the angles? Explain.
41. Open-Ended Choose an angle measure A. Find sin A and cos A. Then use the identities to find cos 2 A 2 eh and sin A 2 . sine , eh over 2 . .

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