AHL 3.10—Compound angle identities Questionbank

Practice AHL 3.10—Compound angle identities 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 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 $\displaystyle \sec x+\tan x=\frac{1+\sin x}{\cos x}=\tan\!\bigl(\tfrac{\pi}{4}+\tfrac{x}{2}\bigr)$ .)
Give your answer to 3 significant figures.

[4]

Let $\phi=\arccos(0.45)$ . 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 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 3

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 $\sec x+\tan x=2.0$ . \nGive your answer to 3 significant figures.

[4]

Let $\phi=\arccos(-0.28)$ . Find exact decimal values of $\sin\phi$ and $\sin(2\phi)$ . \nState the quadrant of $\phi$ and verify that $\cos(\pi-\phi)=-\cos\phi$ .

[4]

A line through the origin has slope $m=\tan\alpha$ . \nFind all $\alpha\in[0,\pi)$ such that $\sec(2\alpha)=1.6$ , and the corresponding slopes $m$ (3 s.f.).

[4]

Question 4

HLPaper 2

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 $\displaystyle \csc x+\cot x=\frac{1+\cos x}{\sin x}=\cot\!\bigl(\tfrac{x}{2}\bigr)$ .)
Give your answer to 3 significant figures.

[4]

Let $\phi=\arccos(0.40)$ . 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 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 5

HLPaper 1

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

Find the values of $\tan x$ and $\cot x$ .

[4]

Solve the given equation for $-\pi \leq x \leq \pi$

[2]

Question 6

HLPaper 1

Show that $\sin

$\sin 15^{\circ}=\frac{\sqrt{6}-\sqrt{2}}{4}$ by using the trigonometric identity for $\sin (A-B)$

[2]

Find a similar expression for $\cos 15^{\circ}$

[2]

Show that $\tan 15^{\circ}=\frac{\sqrt{3}-1}{\sqrt{3}+1}$

[2]

Find a similar expression for $\cot 15^{\circ}$

[2]

Question 7

HLPaper 1

In the diagram below, AD is perpendicular to $\mathrm{BC}, \mathrm{CD}=4, \mathrm{BD}=2$ and $\mathrm{AD}=3$ . Also $\angle \mathrm{CAD}=\alpha$ and $\angle \mathrm{BAD}=\beta$ .

Find the exact value of $\sin (2 \beta)$

[2]

Find the exact value of $\cos (\alpha-\beta)$

[2]

Question 8

HLPaper 1

In the diagram below, AD is perpendicular to $\mathrm{BC}, \mathrm{CD}=4, \mathrm{BD}=2$ and $\mathrm{AD}=3$ . Also $\angle \mathrm{CAD}=\alpha$ and $\angle \mathrm{BAD}=\beta$ .

Find the exact value of $\cos (\alpha+\beta)$

[4]

Find the exact value of $\sin (\alpha+\beta)$

[2]

Question 9

HLPaper 1

Given that ${\text{cos}}\,X = \frac{2}{3}$ and ${\text{sin}}\,Y = \frac{1}{3}$ are acute angles, show that ${\text{cos}}(2X + Y) = - \frac{{2\sqrt{2}}}{27} - \frac{{4\sqrt{5}}}{27}$ .

[7]

Question 10

HLPaper 1

A local artist is creating a mural that incorporates the shape of the arcsine function, $y = \arcsin(x)$ , to symbolize the journey of self-discovery. The artist decides to modify the original function to $y = \arcsin(x - 1) + \frac{\pi}{6}$ to represent a shift in perspective.

Sketch the graph of $y = \arcsin(x - 1) + \frac{\pi}{6}$ . Label key points, domain, range and transformations applied to the original function.

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