Home › Board exam reviewer › Trigonometry

Trigonometry: board exam questions with solutions

On the ECE exam under Mathematics and the REE exam under Mathematics. Try each question before you open its solution, then practise with fresh numbers below: 2 computed question types on this topic.

Loading the reviewer… (it needs JavaScript)

Questions

1. Convert 2.5 rad to degrees.

  1. 143.24°
  2. 150.00°
  3. 45.00°
  4. 71.62°
Show the answer and solution

Answer: A. 143.24°

Degrees = 2.5 × 180/π = 143.24°.

2. What is the exact value of cos 120°?

  1. 1/2
  2. −√3/2
  3. √3/2
  4. −1/2
Show the answer and solution

Answer: D. −1/2

120° is in Quadrant II with a reference angle of 60°. Cosine is negative there, so cos 120° = −cos 60° = −1/2.

3. In triangle ABC, a = 7, b = 8 and the included angle C = 60°. Find side c.

  1. 7.55
  2. 13.00
  3. 10.63
  4. 5.00
Show the answer and solution

Answer: A. 7.55

By the law of cosines, c² = 7² + 8² − 2(7)(8)cos 60° = 49 + 64 − 56 = 57, so c = √57 = 7.55. Using +56 by mistake gives 13.00.

4. In triangle ABC, A = 40°, B = 75° and a = 10 cm. Find side b.

  1. 6.65 cm
  2. 12.50 cm
  3. 15.03 cm
  4. 10.00 cm
Show the answer and solution

Answer: C. 15.03 cm

By the law of sines, b = a·sin B/sin A = 10 × sin 75°/sin 40° = 10 × 0.9659/0.6428 = 15.03 cm. Inverting the ratio gives 6.65 cm.

5. Find the area of a triangle with sides 5 m, 6 m and 7 m.

  1. 15.00 m²
  2. 14.70 m²
  3. 21.00 m²
  4. 10.50 m²
Show the answer and solution

Answer: B. 14.70 m²

By Heron's formula, s = (5 + 6 + 7)/2 = 9 and A = √[9(9 − 5)(9 − 6)(9 − 7)] = √216 = 14.70 m².

6. A 30 m communications tower stands on level ground. What is the angle of elevation of its top from a point 50 m from its base?

  1. 59.04°
  2. 30.96°
  3. 36.87°
  4. 31.00°
Show the answer and solution

Answer: B. 30.96°

tan θ = 30/50 = 0.6, so θ = arctan 0.6 = 30.96°. Using arcsin 0.6 gives 36.87°, and 59.04° is the complement.

7. Which expression is identically equal to sin 2θ?

  1. 2 sin θ
  2. sin²θ − cos²θ
  3. cos²θ − sin²θ
  4. 2 sin θ cos θ
Show the answer and solution

Answer: D. 2 sin θ cos θ

The double-angle identity is sin 2θ = 2 sin θ cos θ. cos²θ − sin²θ equals cos 2θ.

8. Find all solutions of 2cos²θ − cos θ − 1 = 0 for 0° ≤ θ < 360°.

  1. 0°, 60°, 300°
  2. 120°, 240° only
  3. 0°, 120°, 240°
  4. 0°, 180°, 240°
Show the answer and solution

Answer: C. 0°, 120°, 240°

Factor: (2cos θ + 1)(cos θ − 1) = 0, so cos θ = −1/2 or cos θ = 1. cos θ = 1 gives θ = 0°, and cos θ = −1/2 gives θ = 120° and 240°.

Worked examples with computed numbers

These numbers are fixed on this page; in practice they change every time.

9. Two sides of a triangle are 23 m and 9 m, and the angle between them is 57°. How long is the third side, in m?

  1. 24.7
  2. 28.9
  3. 19.61
  4. 27.1
Show the answer and solution

Answer: C. 19.61

Law of cosines: c² = a² + b² − 2ab·cos C = 23² + 9² − 2(23)(9)cos 57°, so c = 19.61.

10. Find the area (cm²) of a circular sector of radius 13 cm and central angle 30°.

  1. 2535
  2. 44.24
  3. 88.49
  4. 6.807
Show the answer and solution

Answer: B. 44.24

A = ½ r² θ with θ in radians: ½ × 13² × 0.5236 = 44.24.

Other topics in the same subjects

Algebra · Analytic geometry · Differential calculus · Integral calculus · Differential equations · Complex numbers · Probability and statistics · Matrices and determinants · Laplace transforms and Fourier series

An independent, free reviewer: not affiliated with the PRC or the Boards. Subject weights and passing rules are from the PRC examination programs and the laws (RA 9292, RA 7920); check the current PRC program before you file. Code questions follow the Philippine Electrical Code 2017 Part 1; check them against your own copy of the Code.