Geometry and Trigonometry practice questions.

Which expression equals sin(A + B)?

• sin A cos B + cos A sin B
• sin A cos B − cos A sin B
• cos A cos B − sin A sin B
• sin A sin B + cos A cos B

Answer: sin A cos B + cos A sin B

Compound-angle identity: sin(A+B) = sin A cos B + cos A sin B. For sine the sign matches the sign in the angle; for cosine it flips: cos(A+B) = cos A cos B − sin A sin B.

What is the angle between the vectors a = (1, 1, 0) and b = (1, 0, 0)?

• 30°
• 45°
• 60°
• 90°

Answer: 45°

cos θ = (a · b)/(|a||b|) = 1/(√2 × 1) = 1/√2, so θ = 45°. Always divide the dot product by both magnitudes.

A plane has normal vector n = (2, −1, 2) and passes through the point (1, 0, 3). Which is its Cartesian equation?

• 2x − y + 2z = 8
• 2x − y + 2z = 0
• x + 3z = 8
• 2x − y + 2z = 6

Answer: 2x − y + 2z = 8

Use r · n = a · n: the right-hand side is 2(1) − 1(0) + 2(3) = 8, giving 2x − y + 2z = 8. Substitute the point back in to verify.

A circle has radius 5 cm. The length of an arc subtending an angle of 1.2 radians at the centre is

• 4.2 cm
• 6 cm
• 7.5 cm
• 12 cm

Answer: 6 cm

With the angle in radians, arc length s = rθ = 5 × 1.2 = 6 cm. This simple formula is a key reason radians are used.

Find the area of a circle with radius 5 (leave your answer in terms of π).

• 10π
• 25π
• 5π
• 50π

Answer: 25π

Area = πr² = π × 5² = 25π.

The period of f(x) = sin(2x) is

• π/2
• π
• 2π
• 4π

Answer: π

For sin(bx) the period is 2π / b. Here b = 2, so the period is 2π / 2 = π.