AHL 3.11—Relationships between trig functions Questionbank

Practice AHL 3.11—Relationships between trig functions with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Evaluate: $\int_0^{\pi} \cos(\pi - x) \, dx$ .

[3]

Question 2

HLPaper 1

Consider the following trigonometric expression

\sin(x+y)-\sin(x-y)

Show that the expression is equal to $2\cos(x)\sin(y)$

[3]

Hence, using mathematical induction and the above identity, prove that $\sum_{r=1}^{n} \cos((2r-1)\alpha) = \frac{\sin(2n\alpha)}{2\sin\alpha}$ for $n \in \mathbb{Z}^+$ .

[8]

Question 3

HLPaper 2

Prove that $1+\tan^2\theta=\sec^2\theta$ and hence show that $(\sec\theta-\tan\theta)(\sec\theta+\tan\theta)=1$

Prove that $1 + \tan^2\theta = \sec^2\theta$ and hence show that $(\sec\theta - \tan\theta)(\sec\theta + \tan\theta) = 1.$

[4]

Solve, for $0\le x<2\pi$ , the equation $\sec x+\tan x=1.7$ . (You may use $\displaystyle \sec x+\tan x=\frac{1+\sin x}{\cos x}=\tan\left(\frac{\pi}{4}+\frac{x}{2}\right)$ .) Give your answer to 3 significant figures.

[4]

Let $\phi=\arccos(0.45)$ . Find the values of $\sin\phi$ and $\sin(2\phi)$ . Give your answers to 3 significant figures. State the quadrant of $\phi$ and verify that $\cos(\pi-\phi)=-\cos\phi$ .

[4]

A line through the origin has equation $y=mx$ and makes an angle $\alpha$ with the positive $x$ -axis, so that $m=\tan\alpha$ . Find all $\alpha\in[0,\pi)$ such that $\sec(2\alpha)=1.5$ , and the corresponding slopes $m$ (to 3 significant figures).

[4]

Question 4

HLPaper 1

Solve the following equation in the interval $[0, 2\pi]$ .

$3 \csc^2 x=4$

[5]

Question 5

HLPaper 2

Prove that $1+\tan^2\theta=\sec^2\theta$ and hence show that

\frac{\sec\theta+\tan\theta}{\sec\theta-\tan\theta}=\frac{1+\sin\theta}{1-\sin\theta}.

Prove $1+\tan^2\theta=\sec^2\theta$ and hence show that

\frac{\sec\theta+\tan\theta}{\sec\theta-\tan\theta}=\frac{1+\sin\theta}{1-\sin\theta}.

[4]

Solve, for $0\le x<2\pi$ , the equation $\sec x+\tan x=2.0$ .

Give your answer to 3 significant figures.

[4]

Let $\phi=\arccos(-0.28)$ . Find exact decimal values of $\sin\phi$ and $\sin(2\phi)$ .

State the quadrant of $\phi$ and verify that $\cos(\pi-\phi)=-\cos\phi$ .

[4]

A line through the origin has slope $m=\tan\alpha$ .

Find all $\alpha\in[0,\pi)$ such that $\sec(2\alpha)=1.6$ , and the corresponding slopes $m$ (3 s.f.).

[4]

Question 6

HLPaper 1

Solve the equation in the interval $[-\pi, \pi]$

$\sin \left(x+\frac{\pi}{3}\right)=2 \cos \left(x-\frac{\pi}{6}\right)$

[5]

Question 7

HLPaper 1

Prove that $\frac{\tan^2(\theta)}{\sec^2(\theta) - 1} = 1$

[3]

Question 8

HLPaper 1

Solve the equation in the interval $[0, \pi]$

$\cos 5 x=\cos 3 x$ ,

[6]

Explain why $\cos (\pi-x)=-\cos x$

[2]

Hence, solve the equation $\cos 5 x+\cos 3 x=0$ , in the interval $[0, \pi]$

[6]

Question 9

HLPaper 2

This question involves trigonometric identities, equations, and properties.

Prove that $1+\cot^2\theta=\csc^2\theta$ and hence show that $(\csc\theta+\cot\theta)(\csc\theta-\cot\theta)=1.$

[4]

Solve, for $0<x<2\pi$ , the equation $\csc x+\cot x=2.3$ .

(You may use $\displaystyle \csc x+\cot x=\frac{1+\cos x}{\sin x}=\cot\left(\tfrac{x}{2}\right)$ . )

Give your answer to 3 significant figures.

[4]

Let $\phi=\arccos(0.40)$ . Find the values of $\sin\phi$ and $\sin(2\phi)$ .

State the quadrant of $\phi$ and verify that $\cos(\pi-\phi)=-\cos\phi$ .

[4]

A line through the origin has equation $y=mx$ and makes an angle $\alpha$ with the positive $x$ -axis, so that $m=\tan\alpha$ .

Find all $\alpha\in[0,\pi)$ such that $\csc(2\alpha)=1.8$ , and the corresponding slopes $m$ (to 3 significant figures).

[4]

Question 10

HLPaper 1

Solve the equation $\sin x = \sin \left(\frac{x}{3}\right)$ in the interval $[0, 4\pi]$ .

Find all values of $x$ satisfying the equation.

[6]