Given $\sin\theta = \frac{3}{5}$ and $\frac{\pi}{2} < \theta < \pi$, find the exact values of $\cos\theta$ and $\tan\theta$.

1. Quadrant 2: cosine and tangent are negative. 2. Pythagorean identity : $\cos^2\theta = 1 - \frac{9}{25} = \frac{16}{25}$. 3. Square root , negative in Q2: $\mathbf{\cos\theta = -\frac{4}{5}}$. 4. Tangent : $\tan\theta = \frac{3/5}{-4/5} = \mathbf{-\frac{3}{4}}$.

Solve $3\tan^2 x - 1 = 0$ for $0^\circ \le x \le 360^\circ$.

1. Rearrange : $\tan^2 x = \frac{1}{3}$. 2. Square root : $\tan x = \pm\frac{1}{\sqrt{3}}$. 3. Reference angle : $30^\circ$. 4. Solutions in all four quadrants : $\mathbf{30^\circ, 150^\circ, 210^\circ, 330^\circ}$.

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IB Maths AA SL · Unit 3: Geometry and Trigonometry

IB Maths AA SL The Unit Circle and Exact Values of Trigonometric Ratios Questions: exact values from the unit circle

Q: How many IB Maths AA SL the unit circle and exact values of trigonometric ratios questions are there?

There are 27 exam-style the unit circle and exact values of trigonometric ratios questions in the AA SL question bank (Paper 1: 25 · Paper 2: 2), graded 6 easy, 13 medium, 5 hard, 3 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Q: Is the unit circle and exact values of trigonometric ratios on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 25 · Paper 2: 2. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Q: Where can I get the mark schemes?

Open the AA SL Unit 3 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

Updated October 2026

Exam-style IB Maths AA SL the unit circle and exact values of trigonometric ratios questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step, showing the working that earns the marks — then try the practice questions and check your working against the mark scheme.

27 questions
Paper 1: 25
Paper 2: 2
6 easy
13 medium
5 hard
3 very hard
3 worked examples

Practise The Unit Circle and Exact Values of Trigonometric Ratios questions →

What you need to know

sin, cos, and tan of 0, 30, 45, 60, 90 degrees (and their radian equivalents) must be memorised for Paper 1. Draw the unit circle in the margin — it saves lives. The unit circle — exact values and identities overview →

sin²θ + cos²θ = 1, plus the double-angle formulas. SL AA Paper 1 asks you to prove or simplify — never to memorise the derivation, but to APPLY it fluently. Trigonometric identities — Pythagorean and double-angle overview →

What's examined in AA SL the unit circle and exact values of trigonometric ratios

The question bank covers these the unit circle and exact values of trigonometric ratios question types (number of questions in brackets):

Unit Circle Exact Values (14)
Solving Trigonometric Equations (7)
Trigonometric Identities (6)

Key formulas

Double-angle: cosine: $\cos 2 θ = \cos^{2} θ - \sin^{2} θ = 2 \cos^{2} θ - 1 = 1 - 2 \sin^{2} θ$
Tangent from sine & cosine: $\tan θ = \frac{\sin θ}{\cos θ}$
Cosine rule: $c^{2} = a^{2} + b^{2} - 2 a b \cos C$
Double-angle: sine: $\sin 2 θ = 2 \sin θ \cos θ$
Sine rule: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
Pythagorean identity: $\sin^{2} θ + \cos^{2} θ = 1$

In the same notation as the IB formula booklet. All AA SL formulas →

The Unit Circle and Exact Values of Trigonometric Ratios worked examples

Worked example 1: Finding exact ratios using quadrants · easy

Given $\sin θ = \frac{3}{5}$ and $\frac{π}{2} < θ < π$ , find the exact values of $\cos θ$ and $\tan θ$ .

Solution

4 steps · 1 to fill in

Examiner tip: A 3-4-5 reference triangle is a valid alternative, provided you remember to apply the ASTC sign rules to the final answer.

Worked example 2: Solving equations with squares · medium

Solve $3 \tan^{2} x - 1 = 0$ for $0^{\circ} \leq x \leq 360^{\circ}$ .

Solution

4 steps

Examiner tip: Forgetting the

\pm

symbol when square-rooting loses half of the solutions. In a full cycle, expect FOUR solutions for a squared trig equation.

Worked example 3: Proving identities algebraically · hard

Prove $\sin 2 x + \cos 2 x - 1 \equiv 2 \sin x (\cos x - \sin x)$ .

Solution

4 steps

Examiner tip: The cosine double angle has three versions; the presence of the

- 1

on the LHS makes

\cos 2 x = 1 - 2 \sin^{2} x

the strategic choice for immediate cancellation.

Try these IB Maths AA SL the unit circle and exact values of trigonometric ratios questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · easy · 5 marks · Paper 1

Using your knowledge of the unit circle, write down the exact values of the following trigonometric ratios:

$\sin (\frac{π}{6})$
$\cos (\frac{3 π}{4})$
$\tan (\frac{4 π}{3})$
$\cos (\frac{3 π}{2})$
$\sin (\frac{11 π}{6})$

Attempt it and see the mark scheme →

Question 2 · medium · 5 marks · Paper 1

Given that $\tan θ = - \frac{4}{3}$ , find the two possible exact values of $\sin θ$ and their corresponding exact values of $\cos θ$ .

Full worked solution Free, no sign-in

Since $\tan θ = - \frac{4}{3}$ is negative, $θ$ must be in Quadrant 2 or Quadrant 4. (R1)
The hypotenuse of the reference triangle is $\sqrt{4^{2} + 3^{2}} = 5$ . (M1)
Case 1 (Quadrant 2): Sine is positive, cosine is negative.
$\sin θ = \frac{4}{5}$ and $\cos θ = - \frac{3}{5}$ . A1A1
Case 2 (Quadrant 4): Sine is negative, cosine is positive.
$\sin θ = - \frac{4}{5}$ and $\cos θ = \frac{3}{5}$ . A1

Every question here has a full worked solution. See them all →

Question 3 · hard · 4 marks · Paper 1

Using the double angle identities, prove the trigonometric identity: $3 \sin 2 θ + \cos 2 θ - 1 = 2 \sin θ (3 \cos θ - \sin θ)$

Attempt it and see the mark scheme →

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All 27 the unit circle and exact values of trigonometric ratios questions with mark schemes →

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FAQ

How many IB Maths AA SL the unit circle and exact values of trigonometric ratios questions are there?

There are 27 exam-style the unit circle and exact values of trigonometric ratios questions in the AA SL question bank (Paper 1: 25 · Paper 2: 2), graded 6 easy, 13 medium, 5 hard, 3 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is the unit circle and exact values of trigonometric ratios on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 25 · Paper 2: 2. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Where can I get the mark schemes?

Open the AA SL Unit 3 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

What else is free here

Save your answers and track your progressA free account remembers the questions you have done on every topic and brings you back to this page.

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Free ready-to-teach lesson (slides and teacher notes): Unit circle & exact values.

IB Maths AA SL The Unit Circle and Exact Values of Trigonometric Ratios Questions: exact values from the unit circle

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What you need to know

What's examined in AA SL the unit circle and exact values of trigonometric ratios

Key formulas

The Unit Circle and Exact Values of Trigonometric Ratios worked examples

Worked example 1: Finding exact ratios using quadrants · easy

Solution

Worked example 2: Solving equations with squares · medium

Solution

Worked example 3: Proving identities algebraically · hard

Solution

Try these IB Maths AA SL the unit circle and exact values of trigonometric ratios questions

Question 1 · easy · 5 marks · Paper 1

Question 2 · medium · 5 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 4 marks · Paper 1

Revise the unit circle and exact values of trigonometric ratios for the exam

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