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AHL 3.10—Compound angle identities - IB Questionbank | RevisionDojo

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 $\displaystyle \sec x+\tan x=\frac{1+\sin x}{\cos x}=\tan\left(\frac{\pi}{4}+\frac{x}{2}\right)$ .) 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 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 $\sec x+\tan x=2.0$ .

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 4

HLPaper 1

In the diagram below, $\mathrm{AD}$ 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 5

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 6

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 7

HLPaper 1

Consider the exact values of trigonometric ratios for $15^{\circ}$ .

Show that $\sin 15^{\circ}=\frac{\sqrt{6}-\sqrt{2}}{4}$ 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 8

HLPaper 1

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

[7]

Question 9

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.

[5]

Question 10

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]

AHL 3.10—Compound angle identities Questionbank