University of the Philippines, ManilaCollege of Arts and Sciences

Department of Physical Sciences and MathematicsMATH 14: Plane Trigonometry

First Departmental ExamJanuary 17, 2012

I. TRUE OR FALSE: Write TRUE if the given statement is accurate; otherwise, write FALSE. (1 pt each)

1. If (a, b, c) is a Pythagorean Triple, then so is (na, nb, nc), where n ∈ N.

2. If α,β are complementary, and β,γ are supplementary, then sin (γ − α) = 1.

3. The function f(x) = sinx cosx is odd.

4. If α and β are coterminal angles, then sin(α− β) = 0.

5. The equation cos(π − θ) cos(θ − π)− sin(π − θ) sin(θ − π) = 1 is an identity.

6. The range of the exponential function y = ex is the set of all nonnegative real numbers.

7. The range of the function f(x) = log(x2 − 6x+ 9

)is the set of real numbers.

8. If sec (θ + π) < 0 and tan(θ + π) > 0, then P (θ) is in the second quadrant.

9. The only fundamental solution to the equation sin θ cos θ tan θ = 1 is θ =π

2.

10. The wrapping function is a function from the set of real numbers to the set of all ordered pairs of real numbers.

II. MULTIPLE CHOICE: Write the letter of the best answer. If you feel that an appropriate answer is notamong the choices, write E. (2 pts each)

1. What is the exact value of 1− sec2 15◦?

A. 2−√3

B. 2 +√3

C. −2−√3

D. −2 +√3

2. Which of the following is not equivalent to sin θ?

A. − sin(θ − π)B. sin(θ − π)

C. − sin(θ + π)

D. sin(π − θ)

3. Which statement below does NOT always hold?

A. If sin θ > 0, then sin θ csc θ > 0.

B. If sin θ < 0, then sin θ tan θ > 0.

C. If tan θ > 0, then tan θ cot θ > 0.

D. If tan θ < 0, then sin θ cos θ < 0.

4. Which of the following equations has an empty solution set?

A. tan θ = cot θ

B. tan θ cot θ = 1

C. sin θ = cos θ

D. sin θ cos θ = 1

5. If tanα = A, the value of tan 2α is:

A.A2

1 +A2

B.A2

1−A2

C.2A

1 +A2

D.2A

1−A2

III. PROBLEM SOLVING: Show your complete solutions for each of the following problem. Write your solutionslegibly and in an organized manner. Deductions may be given for unreadable solutions.

1. Suppose θ is the angle in standard position whose terminal sides coincides with the point P (3, 4). Find:

a. sec(π − θ)

b. cos

(θ

2

) c. sin 2θ

d. tan

(π

4− θ

2

)(2 pts each)

2. Prove the following identities:

a. cos 3θ = 4 cos3 θ − 3 cos θ (5 pts)

b. tan2(θ

2

)= (cot θ − csc θ)2 (5 pts)

c. 2 cosα cosβ cos (α+ β) =cos 2α

2+

cos 2β

2+ cos2(α+ β) (6 pts)

3. Find the fundamental solution set of the following trigonometric equations:

a.1

2(sin θ + cos θ)2 = cos2 2θ + sin2 2θ (5 pts)

b. cos(π + θ) sin(π − θ) tan(π2− θ

)= 1 (6 pts)

c. tan2 x+ 1 = |2 tanx| (5 pts)

4. Solve for x ∈ R: x+ ln 5 = ln(7− 2e2x) (5 pts)

*****END OF EXAM*****Don’t wish it were easier, wish you were better.