A right triangle contains an acute angle $\theta$ such that $\cos\theta = \frac{3}{5}$. Find the exact values of $\sin 2\theta$ and $\cos 2\theta$.

1. Deduce third side: $\sqrt{25-9} = 4$, so $\sin\theta = \frac{4}{5}$. 2. Sine double angle : $\sin 2\theta = 2\sin\theta\cos\theta = 2\cdot\frac{4}{5}\cdot\frac{3}{5} = \mathbf{\frac{24}{25}}$. 3. Cosine double angle : $\cos 2\theta = \cos^2\theta - \sin^2\theta = \frac{9}{25} - \frac{16}{25} = \mathbf{-\frac{7}{25}}$.

An isosceles triangle has two sides of length $a$ and included angle $2\theta$. Prove $\sin 2\theta = 2\sin\theta\cos\theta$ by calculating the area in two different ways.

1. Area (SAS) : $\frac{1}{2}(a)(a)\sin 2\theta = \frac{1}{2}a^2\sin 2\theta$. 2. Half-triangle : base half $= a\sin\theta$, so total base $= 2a\sin\theta$; height $= a\cos\theta$. 3. Area ($\frac{1}{2}bh$) : $\frac{1}{2}(2a\sin\theta)(a\cos\theta) = a^2\sin\theta\cos\theta$. 4. Equate : $\frac{1}{2}a^2\sin 2\theta = a^2\sin\theta\cos\theta$, giving $\mathbf{\sin 2\theta = 2\sin\theta\cos\theta}$. $\blacksquare$

Given that $\sin \theta = \frac{4}{5}$ and $\theta$ is an obtuse angle ($\frac{\pi}{2} < \theta < \pi$), find the exact value of $\sin(2\theta)$.

Use $\cos^2\theta = 1 - \sin^2\theta = 1 - \frac{16}{25} = \frac{9}{25}$ M1 $\theta$ is obtuse, so $\cos\theta < 0$ R1 $\cos\theta = -\frac{3}{5}$ A1 Use $\sin 2\theta = 2\sin\theta\cos\theta = 2\left(\frac{4}{5}\right)\left(-\frac{3}{5}\right)$ M1 $\sin 2\theta = -\frac{24}{25}$ A1

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IB Maths AA SL · Unit 3: Geometry and Trigonometry

Trigonometric Identities (IB Maths AA SL): notes, exam questions and worked examples

Q: Which trig identities are in the AA SL formula booklet?

sin²θ + cos²θ = 1, tan θ = sin θ / cos θ, sin 2θ = 2 sin θ cos θ and the three forms of cos 2θ. Compound angle formulas such as sin(A + B) are not in AA SL.

Q: How do I set out a proof of a trig identity?

Start with one side, apply one identity per line, and finish with the other side. Say which identity you used if it is not obvious.

Q: How many IB Maths AA SL trigonometric identities questions are there?

There are 18 exam-style trigonometric identities questions in the AA SL question bank (Paper 1: 13 · Paper 2: 5), graded 4 easy, 8 medium, 6 hard. Every question has a full IB-style mark scheme (M, A and R marks).

Q: Is trigonometric identities on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 13 · Paper 2: 5. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Updated October 2026 · By Pete Bromfield

The AA SL identities are $\sin^{2} θ + \cos^{2} θ = 1$ , $\tan θ = \frac{\sin θ}{\cos θ}$ and the double-angle formulas for $\sin 2 θ$ and $\cos 2 θ$ , all printed in the formula booklet. You use them to find exact values, to prove that two expressions are equal, and to rewrite an equation before solving it.

18 questions
Paper 1: 13
Paper 2: 5
3 worked examples

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Worked examples
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Trigonometric Identities notes

The Pythagorean identity

$\sin^{2} θ + \cos^{2} θ = 1$ links sin and cos. Given one, find the other and use the quadrant for the sign: if $\sin θ = \frac{3}{5}$ and $θ$ is obtuse, $\cos θ = - \frac{4}{5}$ .

Double-angle formulas

$\sin 2 θ = 2 \sin θ \cos θ$ and $\cos 2 θ = \cos^{2} θ - \sin^{2} θ = 2 \cos^{2} θ - 1 = 1 - 2 \sin^{2} θ$ . If $\sin θ = \frac{3}{5}$ with $θ$ acute, then $\cos θ = \frac{4}{5}$ , $\sin 2 θ = \frac{24}{25}$ and $\cos 2 θ = \frac{7}{25}$ .

Choose the form of $\cos 2 θ$ that matches the rest of the question: $1 - 2 \sin^{2} θ$ if everything else is in $\sin θ$ .

Proving an identity

Start from one side (usually the more complicated one) and work to the other, one justified step per line. Do not work on both sides at once or treat the identity as an equation. For example, $\frac{1 - \cos 2 θ}{\sin 2 θ} = \frac{2 \sin^{2} θ}{2 \sin θ \cos θ} = \tan θ$ .

Exact values

Paper 1 is non-calculator, so answers come out as fractions and surds. Know the exact values of sin, cos and tan at $0, \frac{π}{6}, \frac{π}{4}, \frac{π}{3}, \frac{π}{2}$ , and draw a right-angled triangle when you are given one ratio.

Key formulas

Identities In the booklet: $\tan θ = \frac{\sin θ}{\cos θ}, \cos^{2} θ + \sin^{2} θ = 1$
Double angle In the booklet: $\sin 2 θ = 2 \sin θ \cos θ, \cos 2 θ = \cos^{2} θ - \sin^{2} θ = 2 \cos^{2} θ - 1 = 1 - 2 \sin^{2} θ$
Exact values at $0, \frac{π}{6}, \frac{π}{4}, \frac{π}{3}, \frac{π}{2}$ Learn it: $\sin : 0, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, 1; \cos : reverse order; \tan : 0, \frac{1}{\sqrt{3}}, 1, \sqrt{3}, undefined$
Symmetry Learn it: $\sin (π - θ) = \sin θ, \cos (π - θ) = - \cos θ, \cos (- θ) = \cos θ, \sin (- θ) = - \sin θ$

The IB formula booklet is provided in every exam paper. In the booklet = printed there, so know where to find it; Learn it = not in the booklet, so learn it by heart. All AA SL formulas →

Explained, with a worked example: Trig identities

Trigonometric Identities worked examples

Worked example 1: Double angles from right triangles · easy

A right triangle contains an acute angle $θ$ such that $\cos θ = \frac{3}{5}$ . Find the exact values of $\sin 2 θ$ and $\cos 2 θ$ .

Solution

3 steps

Examiner tip: If the original angle is acute, its double

2 θ

can become obtuse, making

\cos 2 θ

negative. Don't panic if the sign flips.

Worked example 2: Geometric proof of a double angle identity · medium

An isosceles triangle has two sides of length $a$ and included angle $2 θ$ . Prove $\sin 2 θ = 2 \sin θ \cos θ$ by calculating the area in two different ways.

Solution

4 steps

Examiner tip: Geometric proofs require an explicit statement of the formula being used (like Area

= \frac{1}{2} a b \sin C

) BEFORE substituting variables.

Worked example 3: Algebraic proof involving tangent · hard

Given $f (x) = \tan x \sin x + \cos x$ , prove that $f (x) = \frac{1}{\cos x}$ .

Solution

4 steps

Examiner tip: When proving identities mixing

\tan x

with

\sin x

and

\cos x

, immediately substitute

\tan x = \frac{\sin x}{\cos x}

and find a common denominator.

Trigonometric Identities exam-style questions

3 questions from our AA SL bank, written in the style of IB papers, easiest first. One has its full worked solution open here. Each has a full mark scheme in the practice area.

Question 1 · medium · 5 marks · Paper 1

Given that $\sin θ = \frac{4}{5}$ and $θ$ is an obtuse angle ( $\frac{π}{2} < θ < π$ ), find the exact value of $\sin (2 θ)$ .

Full worked solution Free, no sign-in

Use $\cos^{2} θ = 1 - \sin^{2} θ = 1 - \frac{16}{25} = \frac{9}{25}$ M1
$θ$ is obtuse, so $\cos θ < 0$ R1
$\cos θ = - \frac{3}{5}$ A1
Use $\sin 2 θ = 2 \sin θ \cos θ = 2 (\frac{4}{5}) (- \frac{3}{5})$ M1
$\sin 2 θ = - \frac{24}{25}$ A1

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

Question 2 · hard · 5 marks · Paper 1

Given that $\cos x = - \frac{2}{3}$ and that $π < x < \frac{3 π}{2}$ , find the exact value of $\sin (2 x)$ .

Attempt it and see the mark scheme →

Question 3 · hard · 6 marks · Paper 1

Given that $\cos x = \frac{1}{4}$ and $0 < x < \frac{π}{2}$ , find the exact values of:

$\cos (2 x)$
$\sin (2 x)$

Attempt it and see the mark scheme →

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Trigonometric Identities FAQ

Which trig identities are in the AA SL formula booklet?

sin²θ + cos²θ = 1, tan θ = sin θ / cos θ, sin 2θ = 2 sin θ cos θ and the three forms of cos 2θ. Compound angle formulas such as sin(A + B) are not in AA SL.

How do I set out a proof of a trig identity?

Start with one side, apply one identity per line, and finish with the other side. Say which identity you used if it is not obvious.

How many IB Maths AA SL trigonometric identities questions are there?

There are 18 exam-style trigonometric identities questions in the AA SL question bank (Paper 1: 13 · Paper 2: 5), graded 4 easy, 8 medium, 6 hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is trigonometric identities on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 13 · Paper 2: 5. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Next steps

More AA SL Unit 3: Geometry and Right Angled Trigonometry · Sine Rule, Cosine Rule & Area · Radians · The Unit Circle and Exact Values of Trigonometric Ratios · Trigonometric Functions

Teaching this topic? Free ready-to-teach lesson (slides and teacher notes): Trig identities (Pythagorean, double-angle).

Keep revising AA SL: AA SL revision notes · IB Maths AA SL past papers · AA SL common exam mistakes · AA SL syllabus checklist · AA SL geometry and trigonometry knowledge organiser

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Trigonometric Identities (IB Maths AA SL): notes, exam questions and worked examples

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Trigonometric Identities notes

The Pythagorean identity

Double-angle formulas

Proving an identity

Exact values

Key formulas

Trigonometric Identities worked examples

Worked example 1: Double angles from right triangles · easy

Solution

Worked example 2: Geometric proof of a double angle identity · medium

Solution

Worked example 3: Algebraic proof involving tangent · hard

Solution

Trigonometric Identities exam-style questions

Question 1 · medium · 5 marks · Paper 1

Full worked solution Free, no sign-in

Question 2 · hard · 5 marks · Paper 1

Question 3 · hard · 6 marks · Paper 1

Trigonometric Identities FAQ

Next steps

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