Trigonometry — AS & A Level Mathematics (9709) questions by topic

Solve the equation 6 sin θ = 1 + 2/(sin θ) for −180° < θ < 180°.

The diagram shows a sector ABC of a circle with centre A and radius r cm. The angle BAC is α radians, where 0 < α < (1/2)π. It is given that the area of the triangle ABC is 4 cm^2…

The equation of a curve is y = 4 cos 2x + 3 for 0 ≤ x ≤ 2π. State the greatest and least possible values of y.

Prove the identity (tan θ + 7) / (tan^2 θ − 3) ≡ (sin θ cos θ + 7 cos^2 θ) / (1 − 4 cos^2 θ).

Solve the equation 4 sin θ tan θ = 1 + 5 cos θ for −180° < θ < 180°.

Prove that sin^2(2x) + 4 cos^2(x) cos(2x) ≡ 4 cos^4(x). [3]

Express 4 cos θ sin(θ + 30°) in the form R cos(2θ − α) + k, where R 0, 0° < α < 90° and k is a constant.

Express 4 cos θ sin(θ + 30°) in the form R cos(2θ − α) + k, where R 0, 0° < α < 90° and k is a constant.

Express 4 cos θ sin(θ + 30°) in the form R cos(2θ − α) + k, where R 0, 0° < α < 90° and k is a constant.

Express 5 sin(x + (1/6)π) − 4 cos x in the form R sin(x − α), where R 0 and 0 < α < (1/2)π. State the exact value of R and give the value of α correct to 3 decimal places.

Solve the equation 3 cot x - 4 cot 2x = 3 for 0 degrees <= x <= 180 degrees. [6]

Express 7 sin theta + 24 cos theta in the form R cos(theta - alpha), where R 0 and 0 < alpha < (1/2)pi. Give the value of alpha correct to 4 decimal places. [3]

Prove the identity cot²θ − tan²θ ≡ 4 cot 2θ cosec 2θ.

Solve the equation 3 cot θ − 4 cosec^2 θ + 5 = 0 for −π <= θ <= π. [5]

Show that the equation (7 tan θ)/(cos θ) + 12 = 0 can be expressed as 12 sin²θ − 7 sin θ − 12 = 0.

The diagram shows the curve y = k cos(x − (1/6)π) where k is a positive constant and x is measured in radians. The curve crosses the x-axis at point A and B is a minimum point.…

Show that the equation cos θ(7 tan θ − 5 cos θ) = 1 can be written in the form a sin^2 θ + b sin θ + c = 0, where a, b and c are integers to be found.

Show that 3 tan 2θ + tan(θ + 45°) ≡ (tan²θ + 8 tan θ + 1)/(1 − tan²θ). [4]

Prove that 2 sin θ cosec 2θ ≡ sec θ.

Prove that 2 sin θ cosec 2θ ≡ sec θ.

Show that cos^4 theta - sin^4 theta is identically equal to cos 2theta. [3]

Express 3 cos 2x − √3 sin 2x in the form R cos(2x + α), where R 0 and 0 < α < ½π. Give the exact values of R and α.

It is given that β is an angle between 90° and 180° such that sin β = a. Express tan²β − 3 sin β cos β in terms of a.

The diagram shows the curve with equation y = a sin(bx) + c for 0 ≤ x ≤ 2π, where a, b and c are positive constants. State the values of a, b and c.

Find the exact solution of the equation cos(1/6 π) + tan 2x + √3/2 = 0 for −1/4 π < x < 1/4 π.

Solve the equation 4 sin⁴θ + 12 sin²θ − 7 = 0 for 0° ⩽ θ ⩽ 360°.

Prove that cos(θ + 30°) cos(θ + 60°) ≡ (1/4)√3 − (1/2)sin 2θ. [4]

Express 4 sin θ sin(θ + 60°) in the form a + R sin(2θ − α), where a and R are positive integers and 0° < α < 90°.

Prove that cos(θ + 30°) cos(θ + 60°) ≡ ¼√3 − ½ sin 2θ.

Show that sec⁴θ − tan⁴θ ≡ 1 + 2 tan²θ.

Show that the equation tan³x + 2 tan 2x − tan x = 0 may be expressed as tan⁴x − 2 tan²x − 3 = 0 for tan x ≠ 0.

Show that cos⁴θ − sin⁴θ − 4 sin²θ cos²θ ≡ cos²2θ + cos2θ − 1.

Solve the equation 4 sin θ + tan θ = 0 for 0° < θ < 180°.

A curve has equation y = 2 + 3 sin(½x) for 0 ⩽ x ⩽ 4π. State the greatest and least values of y.

By first expanding (cos θ + sin θ)², find the three solutions of the equation (cos θ + sin θ)² = 1 for 0 ≤ θ ≤ π.

Show that the equation 3 tan²x − 3 sin²x − 4 = 0 may be expressed in the form a cos⁴x + b cos²x + c = 0, where a, b and c are constants to be found.

Express 7 cos θ + 24 sin θ in the form R cos(θ − α), where R 0 and 0° < α < 90°. Give the value of α correct to 2 decimal places. [3]

Solve the equation sec²θ + 5tan²θ = 9 + 17secθ for 0° < θ < 360°.

Show that 4 sin(θ + (1/3)π) cos(θ − (1/3)π) ≡ √3 + 2 sin 2θ.

Solve the equation sec²θ + 5 tan²θ = 9 + 17 sec θ for 0° < θ < 360°.

Show that 4 sin(θ + ⅓π) cos(θ − ⅓π) ≡ √3 + 2 sin 2θ.

Show that the equation sin 2θ + cos 2θ = 2 sin² θ can be expressed in the form cos² θ + 2 sin θ cos θ − 3 sin² θ = 0.

Solve the equation 2 cos x − cos ½x = 1 for 0 ≤ x ≤ 2π.

Express 3 cos x + 2 cos(x − 60°) in the form R cos(x − α), where R 0 and 0° < α < 90°. State the exact value of R and give α correct to 2 decimal places.

Show that the equation 4 sin x + 5/(tan x) + 2/(sin x) = 0 may be expressed in the form a cos²x + b cos x + c = 0, where a, b and c are integers to be found.

Find the exact solution of the equation (1/6)π + tan⁻¹(4x) = −cos⁻¹((1/2)√3).

Show that the equation 5 cos θ − sin θ tan θ + 1 = 0 may be expressed in the form a cos²θ + b cos θ + c = 0, where a, b and c are constants to be found.

The diagram shows part of the graph of y = sin(a(x + b)), where a and b are positive constants. State the value of a and one possible value of b.

It is given that θ is an acute angle in degrees such that sin θ = 2/3. Find the exact value of sin(θ + 60°). [3]

Show that cosec θ (3 sin 2θ + 4 sin^3 θ) ≡ 4 + 6 cos θ − 4 cos^2 θ. [3]

Solve the equation sec θ cos(θ − 60°) = 4 for −180° < θ < 180°.

Prove that sin 2x (cot x + 3 tan x) ≡ 4 − 2 cos 2x.

It is given that θ is an acute angle in degrees such that sin θ = 2/3. Find the exact value of sin(θ + 60°).

Show that cosec θ(3 sin 2θ + 4 sin³ θ) ≡ 4 + 6 cos θ − 4 cos² θ.

Given that sin(x + (1/6)π) − sin(x − (1/6)π) = cos(x + (1/3)π) − cos(x − (1/3)π), find the exact value of tan x.

By expressing 3θ as 2θ + θ, prove the identity cos 3θ ≡ 4 cos³θ − 3 cos θ.

Show that the equation cot²θ + 2cos2θ = 4 can be written in the form 4sin⁴θ + 3sin²θ − 1 = 0.