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SL 3.5—Unit circle definitions of sin, cos, tan. Exact trig ratios, ambiguous case of sine rule - IB Questionbank | RevisionDojo

Practice SL 3.5—Unit circle definitions of sin, cos, tan. Exact trig ratios, ambiguous case of sine rule 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

SLPaper 1

Consider the function $f(x) = 3 \sin(kx + \frac{\pi}{4}) - 1$

For the function $f(x)$ find the amplitude of the function.

[1]

Determine the value of $k$ such that the $f(x)$ has a period of $12\pi$

[3]

Find the $x$ -coordinates of the minima and maxima of the function $f(x)$ within a period of $12\pi$ and $k>0$ under the domain $x\in[0,2\pi]$

[6]

Question 2

SLPaper 1

Let $g(x) = \sin(x) + \cos(x)$ .

Find the derivative of the function $g(x)$ .

[1]

Find the equation of the tangent line to the graph of $g(x)$ at $x = \frac{\pi}{4}$ .

[3]

Hence, or otherwise, show that at $x=\frac{\pi}{4}$ the function attains its maximum.

[4]

Question 3

SLPaper 1

The first two terms of an infinite geometric sequence are $v_1 = 18$ and $v_2 = 12\sin^{2} \alpha$ , where $0 < \alpha < 2\pi$ , and $\alpha \neq \pi$ .

Find an expression for $r$ in terms of $\alpha$ .

[2]

Find the values of $\alpha$ which give the greatest value of the sum.

[6]

Question 4

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 5

SLPaper 1

Let $\sin 20^{\circ}=\mathrm{p}$ and $\cos 20^{\circ}=\mathrm{q}$ (so that $\mathrm{p}^{2}+\mathrm{q}^{2}=1$ ). By observing the unit circle express the following in terms of p and/or q .

$\sin 340^{\circ}$

[1]

$\cos 340^{\circ}$

[1]

$\tan 340^{\circ}$

[1]

$\sin \left(-20^{\circ}\right)$

[1]

$\cos \left(-20^{\circ}\right)$

[1]

$\tan \left(-20^{\circ}\right)$

[1]

Question 6

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 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 $\dfrac{3}{5}$ .

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 8

SLPaper 1

Let ABC be the right - angled triangle, where $\angle \mathrm{C}=90^{\circ}$ . The line (AD) bisects $\angle \mathrm{BAC}$ . Here $\mathrm{AC}=5$ and $\mathrm{AD}=6$ .

Find the value of $\cos x$ .

[1]

Find $\cos \mathrm{BAC}$

[2]

Hence find $A B$

[2]

Find $\sin \mathrm{B}$

[1]

Find $\tan \mathrm{BAD}$

[2]

Question 9

SLPaper 1

A right angled triangle has hypotenuse 8 cm . One of its other sides is 5 cm .

Find the exact values for $\sin \theta, \cos \theta$ and $\tan \theta$ , where $\theta$ is the smallest angle in the triangle.

[4]

Question 10

SLPaper 2

The figure shows a plan for a window in the shape of trapezium.

Three sides of the window are 2 m long. The angle between the sloping sides of the window and the base is $\theta$ where $0<\theta<\frac{\pi}{2}$ .

Show that area is $4 \sin \theta+4 \sin \theta \cos \theta$

[4]

Find the possible values of $\theta$ in radians when the area is $5 \mathrm{~m}^{2}$ .

[2]

SL 3.5—Unit circle definitions of sin, cos, tan. Exact trig ratios, ambiguous case of sine rule Questionbank