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AHL 3.8—Unit circle, Pythag identity, solving trig equations graphically - IB Questionbank | RevisionDojo

Practice AHL 3.8—Unit circle, Pythag identity, solving trig equations graphically with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Prove the identity:

\sin ^{2} \theta-\cos ^{2} \theta=-\cos 2 \theta

\sin ^{2} \theta-\cos ^{2} \theta=-\cos 2 \theta

[4]

Question 2

HLPaper 2

Let $P(\cos \theta, \sin \theta)$ be a point on the unit circle, where $\theta$ is measured in radians. The point $Q$ corresponds to the angle $\theta=\frac{7 \pi}{6}$ .

Write down the coordinates of $Q$ .

[2]

Find the value of $\tan \theta$ at $\theta=\frac{7 \pi}{6}$ , giving your answer in exact form.

[2]

Another point $R$ lies on the unit circle such that $\sin \phi=-\frac{1}{2}$ and $\phi \in(0,2 \pi)$ . Find all possible values of $\phi$ .

[3]

Question 3

HLPaper 2

Solve the equation

\tan \theta \cdot \tan (3 \theta)=1,

giving all solutions in the interval $0^{\circ}<\theta<180^{\circ}$ .

\tan \theta \cdot \tan (3 \theta)=1,

[4]

Question 4

HLPaper 2

(i) Prove the identity:

\cos 4 \theta-4 \cos 2 \theta=8 \sin ^{4} \theta-3

Prove the identity:

\cos 4 \theta-4 \cos 2 \theta=8 \sin ^{4} \theta-3

[4]

Hence,

\text { (a) Solve the equation } \cos 4 \theta-4 \cos 2 \theta=1 \text { for }-\frac{\pi}{2}<\theta<\frac{\pi}{2} \text { . }

[3]

Question 5

HLPaper 2

In triangle $\triangle A B C, \angle B A C=58^{\circ}, B C=14.0 \mathrm{~cm}$ , and $A C=15.4 \mathrm{~cm}$ .

Find $\angle A B C, \angle A C B$ , and the length of $A B$ .

[4]

Question 6

HLPaper 1

(i) By expanding $\left(\cos ^{2} x-\sin ^{2} x\right)^{2}$ , or otherwise, show that

\cos ^{4} x+\sin ^{4} x=1-\frac{1}{2} \sin ^{2}(2 x)

By expanding $\left(\cos ^{2} x-\sin ^{2} x\right)^{2}$ , or otherwise, show that

\cos ^{4} x+\sin ^{4} x=1-\frac{1}{2} \sin ^{2}(2 x)

[3]

Hence solve the equation

\cos ^{4} x+\sin ^{4} x=\frac{5}{8}

for $0^{\circ}<x<180^{\circ}$ .

[3]

Question 7

HLPaper 1

In a right triangle, one angle is $30^{\circ}$ and the hypotenuse is 10 cm .

Find the length of the side opposite the $30^{\circ}$ angle.

[2]

Question 8

HLPaper 2

Solve the equation

\cos \theta+3 \cos 2 \theta=2

giving all solutions in the interval $0^{\circ} \leq \theta \leq 180^{\circ}$ .

\cos \theta+3 \cos 2 \theta=2

[5]

Question 9

HLPaper 1

State the coordinates of the point on the unit circle corresponding to the angle $\theta=60^{\circ}$ .

[2]

Question 10

HLPaper 1

A right-angled triangle has legs of 5 cm and 12 cm .

Find the length of the hypotenuse.

[2]

AHL 3.8—Unit circle, Pythag identity, solving trig equations graphically Questionbank