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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 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 3

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 4

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 5

HLPaper 1

Solve the equation

$\arctan x+\arctan 2 x=\arctan \sqrt{2}$

[4]

Question 6

HLPaper 1

Show that $\cos

$\cos (A+B)+\cos (A-B)=2 \cos A \cos B$

[1]

Let $T_{n}(x)=\cos (n \arccos x)$ , where $x$ is a real number, $x \in[-1,1]$ and $n \in Z^{+}$ Find $T_{1}(x)$

[1]

Show that $T_{2}(x)=2 x^{2}-1$

[2]

Use your result in part 1 . to show that $T_{n+1}(x)+T_{n-1}(x)=2 x T_{n}(x)$

[4]

Question 7

HLPaper 2

Let ABC be the right-angled triangle, where $\angle \mathrm{C}=90^{\circ}$ . The line AD bisects $\angle \mathrm{BAC}, \mathrm{BD}=3$ and $\mathrm{DC}=2$ , as shown in the diagram.

Find the exact value of $\tan \angle \mathrm{DAC}$

[4]

Hence find $\angle \mathrm{DAC}$ in degrees.

[2]

Question 8

HLPaper 1

Find the values of $A, B, C$ and $D$ such that

$A=\arctan \frac{1}{3}+\arctan \frac{1}{2}$

$B=\arctan 2+\arctan 3$

$C=\arctan \frac{1}{3}-\arctan 2$

$D=\arctan \frac{2}{3}+\arctan \frac{3}{2}$

[6]

Question 9

HLPaper 1

Let $A, B$ and $C$ be the angles of a triangle. Use the fact $A+B+C=180^{\circ}$ to show that $\tan A+\tan B+\tan C=\tan A \tan B \tan C$

Since $A+B+C=\pi$

[3]

Question 10

HLPaper 2

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

[3]

Solve, for $0\le x<2\pi$ , the equation $\sec x+\tan x=2.4$ .
(You may use $\sec x+\tan x=\dfrac{1+\sin x}{\cos x}=\tan\!\big(\tfrac{\pi}{4}+\tfrac{x}{2}\big)$ .)
Give answers to 3 s.f.

[4]

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

[3]

A line through the origin $y=mx$ makes an angle $\alpha$ with the positive $x$ -axis, so $m=\tan\alpha$ .
Find all $\alpha\in[0,\pi)$ such that $\sec(2\alpha)=1.6$ , and the corresponding slopes $m$ (3 s.f.).

[4]

AHL 3.10—Compound angle identities Questionbank