IB Mathematics Analysis and Approaches Topic SL 3.5—unit Circle Definitions of Sin, Cos, Tan. Exact Trig Ratios, Ambiguous Case of Sine Rule Questionbank (SL/HL)

IB Mathematics Analysis and Approaches Topic SL 3.5—unit Circle Definitions of Sin, Cos, Tan. Exact Trig Ratios, Ambiguous Case of Sine Rule Questionbank (SL/HL) | RevisionDojo

Practice IB Mathematics Analysis and Approaches (AA) Topic SL 3.5—unit Circle Definitions of Sin, Cos, Tan. Exact Trig Ratios, Ambiguous Case of Sine Rule with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for SL 3.5—unit Circle Definitions of Sin, Cos, Tan. Exact Trig Ratios, Ambiguous Case of Sine Rule and mirrors Paper 1, 2, 3 style where relevant.

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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 3.5—unit Circle Definitions of Sin, Cos, Tan. Exact Trig Ratios, Ambiguous Case of Sine Rule Questionbank

IB Mathematics Analysis and Approaches SL 3.5—unit Circle Definitions of Sin, Cos, Tan. Exact Trig Ratios, Ambiguous Case of Sine Rule Questionbank

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IB Mathematics Analysis and Approaches SL 3.5—unit Circle Definitions of Sin, Cos, Tan. Exact Trig Ratios, Ambiguous Case of Sine Rule Questionbank

IB Mathematics Analysis and Approaches SL 3.5—unit Circle Definitions of Sin, Cos, Tan. Exact Trig Ratios, Ambiguous Case of Sine Rule Questionbank

IB Mathematics Analysis and Approaches (AA) Topic SL 3.5—unit Circle Definitions of Sin, Cos, Tan. Exact Trig Ratios, Ambiguous Case of Sine Rule Questionbank

Unlock all 20 questions with a free account