Secondary 4 A-Maths sample questions with answers

Addition and Double Angle Formulae

1. 1Express each of the following as a single trigonometric ratio. (a) sin 52° cos 22° - cos 52° sin 22° (b) 2 sin 4θ cos 4θ (c) cos 3x cos x + sin 3x sin x

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   Answer: (a) sin 30° (b) sin 8θ (c) cos 2x

   1. (a) sin(52° - 22°) = sin 30°
   2. (b) sin 2(4θ) = sin 8θ
   3. (c) cos(3x - x) = cos 2x

2. 2Without using a calculator, find the exact value of cos 75°, in simplest surd form.

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   Answer: (sqrt(6) - sqrt(2)) / 4

   1. cos 75° = cos(45° + 30°) = cos 45° cos 30° - sin 45° sin 30°
   2. = (sqrt(2)/2) × (sqrt(3)/2) - (sqrt(2)/2) × (1/2)
   3. = (sqrt(6) - sqrt(2)) / 4

3. 3Without using a calculator, show that tan 105° = -2 - sqrt(3).

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   Answer: tan 105° = (sqrt(3) + 1) / (1 - sqrt(3)) = -2 - sqrt(3)

   1. tan 105° = tan(60° + 45°) = (tan 60° + tan 45°) / (1 - tan 60° tan 45°)
   2. = (sqrt(3) + 1) / (1 - sqrt(3))
   3. Rationalise with 1 + sqrt(3): (sqrt(3) + 1)(1 + sqrt(3)) / ((1 - sqrt(3))(1 + sqrt(3))) = (4 + 2 sqrt(3)) / (-2)
   4. = -2 - sqrt(3) (shown)

4. 4Given that tan A = 3, find the exact value of tan(A + 45°).

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   Answer: -2

   1. tan(A + 45°) = (tan A + tan 45°) / (1 - tan A tan 45°)
   2. = (3 + 1) / (1 - 3)
   3. = -2

5. 5Show that sin(x + 30°) + cos(x + 60°) ≡ cos x.

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   Answer: The left-hand side simplifies to cos x.

   1. LHS = sin x cos 30° + cos x sin 30° + cos x cos 60° - sin x sin 60°
   2. = (sqrt(3)/2) sin x + (1/2) cos x + (1/2) cos x - (sqrt(3)/2) sin x
   3. = cos x = RHS

6. 6(a) Expand and simplify sin(θ + 60°). (b) Without using a calculator, find the value of cos 50° cos 10° - sin 50° sin 10°. (c) Write (tan 4x + tan x) / (1 - tan 4x tan x) as a single trigonometric ratio.

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   Answer: (a) (1/2) sin θ + (sqrt(3)/2) cos θ (b) 1/2 (c) tan 5x

   1. (a) sin(θ + 60°) = sin θ cos 60° + cos θ sin 60° = (1/2) sin θ + (sqrt(3)/2) cos θ
   2. (b) The pattern cos cos - sin sin is cos(A + B) with A = 50° and B = 10°.
   3. cos 50° cos 10° - sin 50° sin 10° = cos(50° + 10°) = cos 60° = 1/2
   4. (c) The pattern is tan(A + B) with A = 4x and B = x, so the expression is tan(4x + x) = tan 5x.

7. 7Without using a calculator, find the exact value of sin 100° cos 55° - cos 100° sin 55°.

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   Answer: sqrt(2)/2

   1. sin 100° cos 55° - cos 100° sin 55° = sin(100° - 55°)
   2. = sin 45° = sqrt(2)/2

8. 8Without using a calculator, find the exact values of (a) sin 75°, (b) tan 15°.

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   Answer: (a) (sqrt(6) + sqrt(2)) / 4 (b) 2 - sqrt(3)

   1. (a) sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30°
   2. = (sqrt(2)/2) × (sqrt(3)/2) + (sqrt(2)/2) × (1/2) = (sqrt(6) + sqrt(2)) / 4
   3. (b) tan 15° = tan(45° - 30°) = (tan 45° - tan 30°) / (1 + tan 45° tan 30°) = (1 - sqrt(3)/3) ÷ (1 + sqrt(3)/3)
   4. Multiply the numerator and the denominator by 3: (3 - sqrt(3)) / (3 + sqrt(3))
   5. = (3 - sqrt(3))(3 - sqrt(3)) / ((3 + sqrt(3))(3 - sqrt(3))) = (12 - 6 sqrt(3)) / 6
   6. = 2 - sqrt(3)

9. 9Find the exact value of cos(7π/12).

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   Answer: (sqrt(2) - sqrt(6)) / 4

   1. cos(7π/12) = cos(π/3 + π/4) = cos(π/3) cos(π/4) - sin(π/3) sin(π/4)
   2. = (1/2) × (sqrt(2)/2) - (sqrt(3)/2) × (sqrt(2)/2)
   3. = (sqrt(2) - sqrt(6)) / 4

10. 10Given that sin A = 3/5, where A is obtuse, and tan B = 12/5, where 180° < B < 270°, find, without using a calculator, the exact values of (a) cos(A + B), (b) tan(A - B).

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    Answer: (a) 56/65 (b) 63/16

    1. A is in the 2nd quadrant, with a triangle of sides 3, 4 and 5: sin A = 3/5, cos A = -4/5, tan A = -3/4
    2. B is in the 3rd quadrant, with a triangle of sides 5, 12 and 13: sin B = -12/13, cos B = -5/13
    3. (a) cos(A + B) = cos A cos B - sin A sin B = (-4/5)(-5/13) - (3/5)(-12/13)
    4. = 20/65 + 36/65 = 56/65
    5. (b) tan(A - B) = (tan A - tan B) / (1 + tan A tan B) = (-3/4 - 12/5) ÷ (1 + (-3/4)(12/5))
    6. = (-63/20) ÷ (-16/20) = 63/16