SL 3.6—Pythagorean identity, double angles Questionbank

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 $\cos x$ and $\cos 2x$ .

[4]

Given that $|r| \neq 1$ , 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

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 $y$ -coordinate of $P$ is $-\dfrac{2}{\sqrt{5}}$ . Find all possible values of $\cos\theta$ .

[2]

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 $y$ -coordinate of $P$ is $\dfrac{5}{13}$ . Find all possible values of $\cos\theta$ .

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

[2]

Question 6

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 $\dfrac{1}{5}$ . Find all possible values of $\sin\theta$ .

[2]

Question 7

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 $-\dfrac{\sqrt{3}}{2}$ .

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

[2]

If $P$ lies in the third 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 quantity $Q$ measured at $P$ is defined by $Q = (\cos\theta-\sin\theta)^2$ . Determine the maximum and minimum possible values of $Q$ , giving exact values.

[3]

Hence find the range of $\theta$ (in radians, $0\le\theta<2\pi$ ) for which the quantity $Q$ is greater than $1$ .

[3]

Question 8

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 $y$ -coordinate of $P$ is $-\frac{4}{5}$ .

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

[2]

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

[4]

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

[2]

A rotating light beam at $P$ has intensity $I$ given by the equation $I = (\sqrt{5}\sin\theta + \sqrt{5}\cos\theta)^2$ .

Determine the maximum and minimum possible intensities, giving exact values.

[3]

Hence find the range of $\theta$ (in radians, $0 \le \theta < 2\pi$ ) for which the intensity is less than $5$ .

[3]

Question 9

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 $\dfrac{2}{\sqrt{13}}$ .

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

[2]

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

[4]

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

[2]

A quantity $Q$ defined at a general point $P$ on the circle is given by $Q = (\sin\theta + \cos\theta)^2$ .

Determine the maximum and minimum possible values of $Q$ , giving exact values.

[3]

Hence find the range of $\theta$ (in radians, $0 \le \theta < 2\pi$ ) for which the quantity $Q$ is greater than $0$ .

[3]

Question 10

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]