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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

SLPaper 1

Solve $\tan x = 1$ for $x$ in the interval $0 \leq x < 2\pi$ .

[2]

Use a graphing calculator to confirm your solution by sketching the graphs of $y = \tan x$ and $y = 1$ .

[3]

Explain why there are exactly two solutions for $\tan x = 1$ in this interval.

[3]

Describe the behaviour of the tangent function near its asymptotes.

[4]

Question 4

HLPaper 1

Define $\cos \theta$ and $\sin \theta$ in terms of the unit circle.

[3]

Prove the Pythagorean identity:

$\cos^2 \theta + \sin^2 \theta = 1$

[4]

Given that $\sin \theta = \frac{3}{5}$ for some acute angle $\theta$ , find $\cos \theta$ .

[3]

Use the above result to determine $\tan \theta$ .

[2]

Explain the significance of the unit circle in constructing the graphs of ${f(x) = \sin x}$ and ${f(x) = \cos x}$ .

[3]

Question 5

HLPaper 1

Trig Functions and Equations

This question explores the solution of trigonometric equations using both algebraic and graphical methods, along with properties of trigonometric functions such as phase shift and period.

Solve $\sin x = \frac{1}{2}$ for $x$ in the interval $0 \leq x < 2\pi$ .

[3]

Graphically interpret the solutions of $\sin x = \frac{1}{2}$ .

[2]

Find the phase shift and period of the function:

$f(x) = \cos\left(x - \frac{\pi}{4}\right)$

[3]

Verify that $\tan \left(\frac{\pi}{4}\right) = 1$ .

[2]

How can graphical methods help in solving trigonometric equations within a finite interval?

[3]

Question 6

HLPaper 1

The lengths of two of the sides in a triangle are $4$ cm and $5$ cm. Let $\theta$ be the angle between the two given sides. The triangle has an area of $\frac{5\sqrt{15}}{2}$ cm $^2$ .

Show that $\sin\theta = \frac{\sqrt{15}}{4}$ .

[1]

Find the two possible values for the length of the third side.

[6]

Question 7

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 8

SLPaper 2

The depth of water in a port is modelled by the function $d(t) = p\cos qt + 7.5$, for $0 \leqslant t \leqslant 12$, where $t$ is the number of hours after high tide.

At high tide, the depth is 9.7 metres.

At low tide, which is 7 hours later, the depth is 5.3 metres.

Find the value of $p$.

[2]

Find the value of $q$.

[2]

Use the model to find the depth of the water 10 hours after high tide.

[2]

Question 9

HLPaper 2

Consider the trigonometric function $f(x) = 2\sin(x) - \cos(x)$ over the interval $0 \leq x \leq 2\pi$ .

Sketch the graph of $f(x) = 2\sin(x) - \cos(x)$ over the interval $0 \leq x \leq 2\pi$ .

[3]

Using your graph, estimate the solutions to the equation $2\sin(x) - \cos(x) = 0$ within the interval $0 \leq x \leq 2\pi$ .

[3]

Question 10

HLPaper 1

A particle moves along a straight line. Its displacement, $s$ metres, at time $t$ seconds is given by $s = t + \cos 2t, \text{ } t \geqslant 0$ . The first two times when the particle is at rest are denoted by $t_1$ and $t_2$ , where $t_1 < t_2$ .

Find $t_1$ and $t_2$ .

[5]

Find the displacement of the particle when $t = t_1$ .

[2]

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

SLPaper 1

Solve $\tan x = 1$ for $x$ in the interval $0 \leq x < 2\pi$ .

[2]

Use a graphing calculator to confirm your solution by sketching the graphs of $y = \tan x$ and $y = 1$ .

[3]

Explain why there are exactly two solutions for $\tan x = 1$ in this interval.

[3]

Describe the behaviour of the tangent function near its asymptotes.

[4]

Question 4

HLPaper 1

Define $\cos \theta$ and $\sin \theta$ in terms of the unit circle.

[3]

Prove the Pythagorean identity:

$\cos^2 \theta + \sin^2 \theta = 1$

[4]

Given that $\sin \theta = \frac{3}{5}$ for some acute angle $\theta$ , find $\cos \theta$ .

[3]

Use the above result to determine $\tan \theta$ .

[2]

Explain the significance of the unit circle in constructing the graphs of ${f(x) = \sin x}$ and ${f(x) = \cos x}$ .

[3]

Question 5

HLPaper 1

Trig Functions and Equations

This question explores the solution of trigonometric equations using both algebraic and graphical methods, along with properties of trigonometric functions such as phase shift and period.

Solve $\sin x = \frac{1}{2}$ for $x$ in the interval $0 \leq x < 2\pi$ .

[3]

Graphically interpret the solutions of $\sin x = \frac{1}{2}$ .

[2]

Find the phase shift and period of the function:

$f(x) = \cos\left(x - \frac{\pi}{4}\right)$

[3]

Verify that $\tan \left(\frac{\pi}{4}\right) = 1$ .

[2]

How can graphical methods help in solving trigonometric equations within a finite interval?

[3]

Question 6

HLPaper 1

The lengths of two of the sides in a triangle are $4$ cm and $5$ cm. Let $\theta$ be the angle between the two given sides. The triangle has an area of $\frac{5\sqrt{15}}{2}$ cm $^2$ .

Show that $\sin\theta = \frac{\sqrt{15}}{4}$ .

[1]

Find the two possible values for the length of the third side.

[6]

Question 7

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 8

SLPaper 2

The depth of water in a port is modelled by the function $d(t) = p\cos qt + 7.5$, for $0 \leqslant t \leqslant 12$, where $t$ is the number of hours after high tide.

At high tide, the depth is 9.7 metres.

At low tide, which is 7 hours later, the depth is 5.3 metres.

Find the value of $p$.

[2]

Find the value of $q$.

[2]

Use the model to find the depth of the water 10 hours after high tide.

[2]

Question 9

HLPaper 2

Consider the trigonometric function $f(x) = 2\sin(x) - \cos(x)$ over the interval $0 \leq x \leq 2\pi$ .

Sketch the graph of $f(x) = 2\sin(x) - \cos(x)$ over the interval $0 \leq x \leq 2\pi$ .

[3]

Using your graph, estimate the solutions to the equation $2\sin(x) - \cos(x) = 0$ within the interval $0 \leq x \leq 2\pi$ .

[3]

Question 10

HLPaper 1

Find $t_1$ and $t_2$ .

[5]

Find the displacement of the particle when $t = t_1$ .

[2]

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

SLPaper 1

Solve $\tan x = 1$ for $x$ in the interval $0 \leq x < 2\pi$ .

[2]

Use a graphing calculator to confirm your solution by sketching the graphs of $y = \tan x$ and $y = 1$ .

[3]

Explain why there are exactly two solutions for $\tan x = 1$ in this interval.

[3]

Describe the behaviour of the tangent function near its asymptotes.

[4]

Question 4

HLPaper 1

Define $\cos \theta$ and $\sin \theta$ in terms of the unit circle.

[3]

Prove the Pythagorean identity:

$\cos^2 \theta + \sin^2 \theta = 1$

[4]

Given that $\sin \theta = \frac{3}{5}$ for some acute angle $\theta$ , find $\cos \theta$ .

[3]

Use the above result to determine $\tan \theta$ .

[2]

Explain the significance of the unit circle in constructing the graphs of ${f(x) = \sin x}$ and ${f(x) = \cos x}$ .

[3]

Question 5

HLPaper 1

Trig Functions and Equations

This question explores the solution of trigonometric equations using both algebraic and graphical methods, along with properties of trigonometric functions such as phase shift and period.

Solve $\sin x = \frac{1}{2}$ for $x$ in the interval $0 \leq x < 2\pi$ .

[3]

Graphically interpret the solutions of $\sin x = \frac{1}{2}$ .

[2]

Find the phase shift and period of the function:

$f(x) = \cos\left(x - \frac{\pi}{4}\right)$

[3]

Verify that $\tan \left(\frac{\pi}{4}\right) = 1$ .

[2]

How can graphical methods help in solving trigonometric equations within a finite interval?

[3]

Question 6

HLPaper 1

The lengths of two of the sides in a triangle are $4$ cm and $5$ cm. Let $\theta$ be the angle between the two given sides. The triangle has an area of $\frac{5\sqrt{15}}{2}$ cm $^2$ .

Show that $\sin\theta = \frac{\sqrt{15}}{4}$ .

[1]

Find the two possible values for the length of the third side.

[6]

Question 7

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 8

SLPaper 2

The depth of water in a port is modelled by the function $d(t) = p\cos qt + 7.5$, for $0 \leqslant t \leqslant 12$, where $t$ is the number of hours after high tide.

At high tide, the depth is 9.7 metres.

At low tide, which is 7 hours later, the depth is 5.3 metres.

Find the value of $p$.

[2]

Find the value of $q$.

[2]

Use the model to find the depth of the water 10 hours after high tide.

[2]

Question 9

HLPaper 2

Consider the trigonometric function $f(x) = 2\sin(x) - \cos(x)$ over the interval $0 \leq x \leq 2\pi$ .

Sketch the graph of $f(x) = 2\sin(x) - \cos(x)$ over the interval $0 \leq x \leq 2\pi$ .

[3]

Using your graph, estimate the solutions to the equation $2\sin(x) - \cos(x) = 0$ within the interval $0 \leq x \leq 2\pi$ .

[3]

Question 10

HLPaper 1

Find $t_1$ and $t_2$ .

[5]

Find the displacement of the particle when $t = t_1$ .

[2]

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

Trig Functions and Equations

Trig Functions and Equations

Trig Functions and Equations