IB Mathematics Analysis and Approaches Topic SL 3.6—pythagorean Identity, Double Angles Questionbank (SL/HL)

IB Mathematics Analysis and Approaches Topic SL 3.6—pythagorean Identity, Double Angles Questionbank (SL/HL) | RevisionDojo

Practice IB Mathematics Analysis and Approaches (AA) Topic SL 3.6—pythagorean Identity, Double Angles with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for SL 3.6—pythagorean Identity, Double Angles and mirrors Paper 1, 2, 3 style where relevant.

Get instant solutions, detailed explanations, and build confidence with questions aligned to IB examiner expectations.

Paper

Level

Question 1

SLPaper 1

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

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

[3]

Hence, or otherwise, show that $sin (2 θ) = \frac{2 t a n θ}{1 + t a n ^{2} θ} .$

[3]

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

[4]

Question 2

SLPaper 1

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

At a certain instant, the $y$ -coordinate of $P$ is $\frac{5}{13}$ . Find all possible values of $cos θ$ .

Find all possible values of $cos θ$ .

[2]

Question 3

HLPaper 1

A point $P$ moves around the unit circle centered at the origin $O (0, 0)$ with coordinates $P (cos θ, sin θ)$ . The $y$ -coordinate of $P$ is $\frac{12}{13}$ .

Find all possible values of $cos θ$ .

[2]

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

[4]

Verify that the identity $1 + tan^{2} (2 θ) = sec^{2} (2 θ)$ holds for your values.

[2]

Now let $θ$ vary with $0 \leq θ < 2 π$ . Let $Q = (2 sin θ - 2 cos θ)^{2}$ . Determine the maximum and minimum possible values of $Q$ , giving exact values.

[3]

Hence find the range of $θ$ (in radians, $0 \leq θ < 2 π$ ) for which the measured quantity $Q$ is greater than $3$ .

[3]

Question 4

SLPaper 1

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

The $y$ -coordinate of $P$ is $- \frac{1}{5}$ . Find all possible values of $cos θ$ .

[2]

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

[4]

Verify that the identity $1 + tan^{2} (2 θ) = sec^{2} (2 θ)$ holds for your values.

[2]

A rotating sensor at $P$ measures a quantity $Q$ given by $(5 cos θ - 5 sin θ)^{2}$ . Determine the maximum and minimum possible values of $Q$ , giving exact values.

[3]

Hence find the range of $θ$ (in radians, $0 \leq θ < 2 π$ ) for which the measured quantity $Q$ is less than $10$ .

[3]

Question 5

SLPaper 1

A point $P$ moves around the unit circle centered at the origin $O (0, 0)$ with coordinates $P (cos θ, sin θ)$ , where $θ$ is measured in radians.

At a certain instant, the $x$ -coordinate of $P$ is $- \frac{3}{3}$ . Find all possible values of $sin θ$ .

[2]

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

[4]

Verify that the identity $cot^{2} (2 θ) + 1 = csc^{2} (2 θ)$ holds for your values.

[2]

A rotating beacon at $P$ emits a light beam of intensity $I$ , where $I = (sin θ + cos θ)^{2}$ . Determine the maximum and minimum possible intensities, giving exact values.

[3]

Hence find the range of $θ$ (in radians, $0 \leq θ < 2 π$ ) for which the intensity is less than $1$ .

[3]

IB Mathematics Analysis and Approaches SL 3.6—pythagorean Identity, Double Angles Questionbank

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IB Mathematics Analysis and Approaches SL 3.6—pythagorean Identity, Double Angles Questionbank

IB Mathematics Analysis and Approaches (AA) Topic SL 3.6—pythagorean Identity, Double Angles Questionbank

Unlock all 22 questions with a free account