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SL 3.6—Pythagorean identity, double angles - IB Questionbank | RevisionDojo

Practice SL 3.6—Pythagorean identity, double angles 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

SL & HLPaper 1

Consider the expansion of $(\sin x + \cos x)^4$ .

Using the binomial expansion, find the 2nd and 4th terms ( $T_2,T_4$ respectively) in ascending powers of $\sin x$

[3]

Hence show that $T_2-T_4=\sin(4x)$

[5]

Question 2

SL & HLPaper 1

Consider a geometric sequence where the first term $a_1 = 4\sin x$ and the second term $a_2 = \sin 4x$ , where $\sin x \neq 0$ .

Find the common ratio $r$ of this geometric sequence in terms of $\cos$

[4]

Given that $|r| \not= 1$ , explain why the sum to infinity exists and find it for $x=\frac{\pi}{6}$ giving your answer in the form $\frac{a+b\sqrt{c}}{d}$

[5]

Question 3

SLPaper 1

Prove the identity
$\frac{1-\tan^2\theta}{1+\tan^2\theta}=\cos(2\theta),$ for all values of $\theta$ for which both sides are defined.

Start from the left-hand side and express everything in terms of $\sin\theta$ and $\cos\theta$ .

[3]

Hence, or otherwise, show that $\sin(2\theta)=\frac{2\tan\theta}{1+\tan^2\theta}.$

[3]

If $\tan\theta=\dfrac{3}{4}$ , find the exact values of $\sin(2\theta)$ and $\cos(2\theta)$ .

[4]

Question 4

SLPaper 1

By using the double angle identities for $\cos 2 \theta$ .

Show that $\sin 15^{\circ}=\frac{\sqrt{2-\sqrt{3}}}{2}$

[4]

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

[4]

Question 5

SLPaper 1

A point $P$ moves around the unit circle centered at the origin $O(0,0)$ with coordinates $P(\cos\theta,\sin\theta)$ .

At a certain instant, the $x$ -coordinate of $P$ is $\dfrac{3}{5}$ .

Find all possible values of $\sin\theta$ .

[2]

If $P$ lies in the second quadrant, find the exact values of $\sin(2\theta)$ , $\cos(2\theta)$ , and $\tan(2\theta)$ .

[4]

Verify that the identity $1+\tan^2(2\theta)=\sec^2(2\theta)$ holds for your values.

[2]

A rotating beacon at the point $P$ emits a light beam of intensity proportional to $(\sin\theta+\cos\theta)^2$ .
As $\theta$ increases from $0$ to $2\pi$ , determine the maximum and minimum possible intensities, giving exact values.

[4]

Hence find the range of $\theta$ (in radians, $0\le\theta<2\pi$ ) for which the intensity exceeds 1.5.

[3]

Question 6

SLPaper 1

Let $f(x)=\sin ^{3} x+\cos ^{3} x \tan x$ where $\frac{\pi}{2}<x<\pi$ .

Show that $f(x)=\sin x$

[3]

Let $\sin x=\frac{2}{3}$ . Find the exact value of $f(2 x)$

[3]

Question 7

SLPaper 1

Given that $\sin x=\mathrm{p}$ , where $x$ is an acute angle. Find the value of the following in terms of p .

$\cos x$ and $\tan x$
$\cos 2 x$
$\sin 2 x$
$\tan 2 x$
$\sin 4 x$

[6]

Question 8

SLPaper 2

If A is an obtuse angle in a triangle and $\sin \mathrm{A}=\frac{5}{13}$ .

Find the value of $\sin 2 \mathrm{~A}$ and $\cos 2 \mathrm{~A}$ .

[4]

Question 9

SLPaper 1

Let $\theta$ be the angle for which $\cos \theta+\sin \theta=\frac{4}{3}$ . To find :

$\sin 2 \theta$
$\cos 4 \theta$

[6]

Question 10

SLPaper 1

Given that $\sin x=\frac{1}{3}$ , where $x$ is an acute angle. Find the exact value of:

$\cos x$
$\cos 2 x$ [3]
$\sin 2 x$

[6]

SL 3.6—Pythagorean identity, double angles Questionbank