Trigonometry MCQs With Answers
A handful of standard values and identities is enough to answer most trigonometry questions in competitive tests. These trigonometry MCQs cover the values of sin, cos and tan at 0°, 30°, 45°, 60° and 90°, the identities sin²θ + cos²θ = 1 and sec²θ − tan²θ = 1, finding one ratio from another, and heights and distances using shadows and angles of elevation. They are useful for NTS, entry tests, PPSC and FPSC lecturer and teaching jobs, and matric and intermediate students.
Take a scored quiz instead →The value of sin 90° is:
Standard trigonometric value.
Find the value of tan 45°.
Standard trigonometric value.
Find the value of tan 0°.
Trig value.
Find the value of sin 30°.
Trigonometric value.
Value of cos 90° is:
Trig value.
Find the value of tan 45° + cot 45°.
1 + 1 = 2.
Value of sin 0° is:
Trig value.
Value of cos 60° is:
Trig value.
The value of (sin^2 θ + cos^2 θ) is always:
Trigonometric identity.
Value of cos 0°.
Trig value.
Find the value of (sin 30°)^2 + (cos 30°)^2.
Trig identity.
Find the value of sin 30° + cos 60°.
0.5 + 0.5 = 1.
If sin(theta) = 3/5, cos(theta) = ?
3-4-5 triangle. cos=4/5.
If sin(theta) = 4/5, find tan(theta).
sin=4/5 (O/H). Adj = sqrt(5^2-4^2) = 3. Tan = O/A = 4/3.
If sin(theta) = 1/2, what is theta?
sin(30°) = 0.5.
If tan(theta) = 1, what is theta?
tan(45°) = 1.
If sin(theta) = 0.6, find cos(theta).
cos = sqrt(1 - 0.6^2) = sqrt(0.64) = 0.8.
Evaluate sin(60°)cos(30°) + cos(60°)sin(30°).
- The expression matches the identity sin A cos B + cos A sin B = sin(A + B).
- So it equals sin(60° + 30°) = sin 90° = 1.
- The long route agrees: (√3/2)(√3/2) + (1/2)(1/2) = 3/4 + 1/4 = 1.
- Option D √3 appears if the second term is dropped, so both products must be added.
- The companion identity is cos A cos B − sin A sin B = cos(A + B), which is worth learning next to this one.
If tan(theta) = 4/3, find the value of (3 sin(theta) + 2 cos(theta)) / (3 sin(theta) – 2 cos(theta)).
Divide numerator and denominator by cos(theta): (3 tan(theta) + 2) / (3 tan(theta) - 2) = (3(4/3) + 2) / (3(4/3) - 2) = (4 + 2) / (4 - 2) = 6 / 2 = 3.
If sin(theta) = 5/13, find the value of tan(theta) + sec(theta).
Opposite = 5, Hypotenuse = 13 => Adjacent = sqrt(169 - 25) = 12. tan(theta) = 5/12, sec(theta) = 13/12. Sum = (5 + 13) / 12 = 18 / 12 = 1.5.
If cos(theta) = 4/5, find the value of (1 – tan(theta)) / (1 + tan(theta)).
cos(theta) = 4/5 => sin(theta) = 3/5 => tan(theta) = 3/4. Expression = (1 - 3/4) / (1 + 3/4) = (1/4) / (7/4) = 1/7.
If tan(theta) = 3/4, find the value of sin(theta) + cos(theta).
Opposite = 3, Adjacent = 4 => Hypotenuse = 5. sin(theta) = 3/5, cos(theta) = 4/5. Sum = 3/5 + 4/5 = 7/5.
Evaluate: cos(0°) + sin(90°) + tan(45°).
cos(0°) = 1, sin(90°) = 1, tan(45°) = 1. Sum = 1 + 1 + 1 = 3.
If a post of height 6 m casts a shadow 2*sqrt(3) m long, what is the angle of elevation of the sun?
tan(theta) = Height / Shadow = 6 / (2*sqrt(3)) = 3 / sqrt(3) = sqrt(3). theta = arctan(sqrt(3)) = 60°.
What is the value of sin^2(30°) + cos^2(60°)?
sin(30°) = 1/2 => sin^2(30°) = 1/4. cos(60°) = 1/2 => cos^2(60°) = 1/4. Sum = 1/4 + 1/4 = 2/4 = 1/2.
If sec(theta) + tan(theta) = 3, what is the value of sec(theta) – tan(theta)?
Since sec^2(theta) - tan^2(theta) = 1 => (sec(theta) + tan(theta))(sec(theta) - tan(theta)) = 1. So sec(theta) - tan(theta) = 1 / 3.
If cos(theta) = 3/5, what is the value of (1 + tan^2(theta))?
1 + tan^2(theta) = sec^2(theta) = 1 / cos^2(theta) = 1 / (3/5)^2 = 1 / (9/25) = 25/9.
A vertical pole 12 m long casts a shadow 8 m long on the ground. At the same time, a tower casts a shadow 40 m long. Find the height of the tower.
By similar triangles: Height1 / Shadow1 = Height2 / Shadow2 => 12 / 8 = H / 40 => 1.5 = H / 40 => H = 60 m.
What is the angle of elevation of the sun when the length of the shadow of a tree is equal to its height?
tan(theta) = Height / Shadow. Since Height = Shadow, tan(theta) = 1 => theta = 45°.
A pole cast a shadow 10 m long when the elevation of the sun is 45°. What is the height of the pole?
tan(45°) = Height / Shadow => 1 = Height / 10 => Height = 10 m.
1 radian is approximately equal to:
Since π radians = 180°, 1 radian = 180°/π ≈ 180/3.1416 ≈ 57.296°, i.e. about 57.3° (57°17′45″).
Convert 5π/6 radians into degrees.
To convert radians to degrees, multiply by 180°/π: (5π/6) × (180°/π) = 5 × 30° = 150°.
What is the value of sin 45° × cos 45°?
sin 45° = cos 45° = 1/√2, so the product = (1/√2) × (1/√2) = 1/2.
1 + cot²θ is equal to:
Divide the identity sin²θ + cos²θ = 1 by sin²θ: 1 + cos²θ/sin²θ = 1/sin²θ, i.e. 1 + cot²θ = cosec²θ.
sin(90° − θ) is equal to:
Sine and cosine are co-functions: the sine of an angle equals the cosine of its complement. Therefore sin(90° − θ) = cos θ.
What is the value of sin 120°?
120° = 180° − 60° lies in the second quadrant, where sine is positive. So sin 120° = sin 60° = √3/2.
What is the value of cos 180°?
On the unit circle, the angle 180° corresponds to the point (−1, 0). Cosine is the x-coordinate, so cos 180° = −1, while sin 180° = 0.
The value of tan 90° is:
tan θ = sin θ / cos θ. At 90°, sin 90° = 1 and cos 90° = 0, so tan 90° = 1/0, which is not defined. As θ approaches 90°, tan θ grows without limit.
In which quadrant are both sin θ and cos θ negative?
In the third quadrant (180° to 270°) both the x- and y-coordinates are negative, so cos θ and sin θ are both negative, while their ratio tan θ is positive.
Which of the following is equal to sin 2θ?
From sin(A + B) = sin A cos B + cos A sin B, putting A = B = θ gives sin 2θ = 2 sin θ cos θ.
cos 2θ is equal to:
Using cos(A + B) = cos A cos B − sin A sin B with A = B = θ gives cos 2θ = cos²θ − sin²θ. Using sin²θ + cos²θ = 1, it can also be written as 2cos²θ − 1 or 1 − 2sin²θ.
What is the period of the function tan x?
tan(x + π) = tan x for every x where it is defined, and no smaller positive number has this property, so the period of tan x is π (180°).
The value of sin 15° is:
sin 15° = sin(45° − 30°) = sin 45° cos 30° − cos 45° sin 30° = (1/√2)(√3/2) − (1/√2)(1/2) = (√3 − 1)/(2√2) ≈ 0.2588.
A kite is flying on a straight string 100 m long that makes an angle of 30° with the ground. How high is the kite?
The string is the hypotenuse and the height is the side opposite the 30° angle. Height = 100 × sin 30° = 100 × 1/2 = 50 m.
From a point on level ground 30 m from the foot of a tower, the angle of elevation of its top is 30°. What is the height of the tower?
tan 30° = height / distance, so height = 30 × tan 30° = 30 × 1/√3 = 10√3 m ≈ 17.32 m.
From the top of a 60 m high tower, the angle of depression of a car on the level road is 30°. How far is the car from the foot of the tower?
The angle of depression from the top equals the angle of elevation from the car (alternate angles). So tan 30° = 60/d, giving d = 60 ÷ (1/√3) = 60√3 m ≈ 103.9 m.
Two sides of a triangle are 8 cm and 10 cm, and the angle between them is 30°. What is the area of the triangle?
Area = ½ × a × b × sin C = ½ × 8 × 10 × sin 30° = 40 × 1/2 = 20 cm².
In any triangle ABC, according to the law of sines, a / sin A is equal to:
The law (rule) of sines states that a/sin A = b/sin B = c/sin C = 2R, where a, b and c are the sides opposite angles A, B and C, and R is the radius of the circumcircle.
What is the maximum value of 3 sin θ + 4 cos θ?
For a sin θ + b cos θ, the maximum value is √(a² + b²) and the minimum is −√(a² + b²). Here √(3² + 4²) = √25 = 5.
If sin A = 3/5 and A is an acute angle, what is the value of sin 2A?
Since A is acute, cos A = √(1 − 9/25) = 4/5. Then sin 2A = 2 sin A cos A = 2 × 3/5 × 4/5 = 24/25.