Exact Trigonometric Values: 30°, 45°, 60° and Related Angles

Introduction

In CAIE A-Level Mathematics (9709), a large number of trigonometry questions — from solving equations to proving identities and integrating trigonometric functions — require you to evaluate expressions without a calculator. The syllabus explicitly demands that you know the exact values of $\sin$ , $\cos$ , and $\tan$ for $30°$ , $45°$ , $60°$ , and any related angle obtained by reflection or rotation into other quadrants. Exact values keep answers in surd form, which is always required unless a decimal approximation is explicitly asked for. Memorising these values (and knowing how to derive them) earns marks efficiently throughout the paper.

Core Concept

Deriving the values geometrically

45° — the isosceles right-angled triangle

Take a right-angled isosceles triangle with the two equal legs each of length $1$ . By Pythagoras, the hypotenuse is $\sqrt{2}$ . Both acute angles are $45°$ .

30° and 60° — the equilateral triangle

Take an equilateral triangle with side length $2$ . Drop a perpendicular from one vertex to the opposite side. This bisects the base (giving $1$ ) and the apex angle. The perpendicular has length $\sqrt{3}$ (by Pythagoras: $2^2 - 1^2 = 3$ ). The two triangles formed each have angles $30°$ , $60°$ , $90°$ .

Reading off from these two triangles gives all six values below.

Related angles and the CAST diagram

For angles beyond $0°$ – $90°$ , use the CAST rule: only the labelled ratios are positive in each quadrant.

First quadrant ( $0°$ – $90°$ ): All positive.
Second quadrant ( $90°$ – $180°$ ): Sine positive.
Third quadrant ( $180°$ – $270°$ ): Tangent positive.
Fourth quadrant ( $270°$ – $360°$ ): Cosine positive.

The reference angle (the acute angle to the nearest part of the $x$ -axis) is used to find the magnitude; the quadrant determines the sign.

Angle $\theta$	Related angle	Rule
$180° - \theta$	$\theta$	$\sin(180°-\theta)=\sin\theta$ , $\cos(180°-\theta)=-\cos\theta$
$180° + \theta$	$\theta$	$\sin(180°+\theta)=-\sin\theta$ , $\cos(180°+\theta)=-\cos\theta$
$360° - \theta$	$\theta$	$\sin(360°-\theta)=-\sin\theta$ , $\cos(360°-\theta)=\cos\theta$

Negative angles follow the same pattern: $\sin(-\theta) = -\sin\theta$ , $\cos(-\theta) = \cos\theta$ .

Key Formulae & Definitions

The fundamental exact values

Angle	$\sin$	$\cos$	$\tan$
$30°$	$\dfrac{1}{2}$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{1}{\sqrt{3}} = \dfrac{\sqrt{3}}{3}$
$45°$	$\dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{2}}{2}$	$\dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{2}}{2}$	$1$
$60°$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{1}{2}$	$\sqrt{3}$

Memory tip: For sine, the values $\frac{1}{2},\ \frac{\sqrt{2}}{2},\ \frac{\sqrt{3}}{2}$ increase with the angle. Cosine is the reverse. Tangent is always $\sin\theta \div \cos\theta$ .

Boundary (quadrantal) values for completeness

Angle	$\sin$	$\cos$	$\tan$
$0°$	$0$	$1$	$0$
$90°$	$1$	$0$	undefined
$180°$	$0$	$-1$	$0$
$270°$	$-1$	$0$	undefined

Worked Examples

Example 1 — Finding exact values of related angles

Find the exact value of $\cos 150°$ and $\tan 240°$ .

Step 1 — Identify the reference angle.

$150° = 180° - 30°$ , so the reference angle is $30°$ . $240° = 180° + 60°$ , so the reference angle is $60°$ .

Step 2 — Determine the quadrant and sign.

$150°$ lies in the second quadrant: cosine is negative. $240°$ lies in the third quadrant: tangent is positive.

Step 3 — Apply the sign.

\cos 150° = -\cos 30° = -\frac{\sqrt{3}}{2}

\tan 240° = +\tan 60° = \sqrt{3}

Example 2 — Evaluating a trigonometric expression exactly

Without a calculator, evaluate $\sin^2 120° + \cos^2 150° - \tan 135°$ .

Step 1 — Find each exact value.

$120° = 180° - 60°$ → second quadrant → $\sin 120° = +\sin 60° = \dfrac{\sqrt{3}}{2}$

$150° = 180° - 30°$ → second quadrant → $\cos 150° = -\cos 30° = -\dfrac{\sqrt{3}}{2}$

$135° = 180° - 45°$ → second quadrant → $\tan 135° = -\tan 45° = -1$

Step 2 — Substitute.

\sin^2 120° + \cos^2 150° - \tan 135°

= \left(\frac{\sqrt{3}}{2}\right)^2 + \left(-\frac{\sqrt{3}}{2}\right)^2 - (-1)

Step 3 — Simplify.

= \frac{3}{4} + \frac{3}{4} + 1 = \frac{3}{2} + 1 = \frac{5}{2}

Example 3 — Solving a trigonometric equation using exact values

Solve $\cos\theta = -\dfrac{1}{2}$ for $0° \leq \theta \leq 360°$ .

Step 1 — Recognise the exact value.

$\cos 60° = \dfrac{1}{2}$ , so the reference angle is $60°$ .

Step 2 — Cosine is negative in the second and third quadrants.

\theta = 180° - 60° = 120° \quad \text{or} \quad \theta = 180° + 60° = 240°

Answer: $\theta = 120°$ or $\theta = 240°$ .

Common Mistakes & Examiner Pitfalls

Swapping sin and cos for 30° and 60°. Remember: $\sin 30° = \frac{1}{2}$ (small angle, small value) and $\sin 60° = \frac{\sqrt{3}}{2}$ (larger angle, larger value). Cosine is the opposite order.
Forgetting to rationalise the denominator. Examiners expect $\dfrac{\sqrt{3}}{3}$ or $\dfrac{\sqrt{2}}{2}$ , not $\dfrac{1}{\sqrt{3}}$ or $\dfrac{1}{\sqrt{2}}$ as a final answer.
Incorrect sign in a related quadrant. Always identify the quadrant first, then assign the sign. A common error is writing $\cos 120° = +\frac{1}{2}$ (forgetting the negative in Q2).
Using $\tan 90°$ or $\tan 270°$ . These are undefined — do not attempt to assign a value.
Mixing degrees and radians. If the question uses radians (e.g. $\frac{\pi}{6}$ , $\frac{\pi}{4}$ , $\frac{\pi}{3}$ ), the exact values are identical in magnitude; apply the same CAST reasoning.
Missing solutions in a given interval. When solving equations, always check all relevant quadrants, not just the principal value.

Practice Questions

Q1. Write down the exact value of $\sin 300°$ .

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$300° = 360° - 60°$ , so the reference angle is $60°$ . $300°$ is in the fourth quadrant: sine is negative.

\sin 300° = -\sin 60° = -\frac{\sqrt{3}}{2}

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Q2. Find the exact value of $\tan^2 30° \times \cos 60°$ .

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\tan 30° = \frac{1}{\sqrt{3}}, \quad \cos 60° = \frac{1}{2}

\tan^2 30° \times \cos 60° = \left(\frac{1}{\sqrt{3}}\right)^2 \times \frac{1}{2} = \frac{1}{3} \times \frac{1}{2} = \frac{1}{6}

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Q3. Solve $\tan\theta = -\sqrt{3}$ for $0° \leq \theta \leq 360°$ .

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Reference angle: $\tan 60° = \sqrt{3}$ , so reference angle $= 60°$ .

Tangent is negative in the second and fourth quadrants.

\theta = 180° - 60° = 120° \quad \text{or} \quad \theta = 360° - 60° = 300°

Answer: $\theta = 120°$ or $\theta = 300°$ .

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Q4. Without a calculator, show that $\sin 45° \cos 30° + \cos 45° \sin 30° = \dfrac{\sqrt{6}+\sqrt{2}}{4}$ .

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\sin 45° \cos 30° + \cos 45° \sin 30°

= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}

= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6}+\sqrt{2}}{4} \quad \checkmark

(Note: this is the compound angle formula for $\sin 75°$ — a useful connection.)

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Q5. Find all values of $\theta$ in the interval $-180° \leq \theta \leq 180°$ satisfying $\sin\theta = \dfrac{\sqrt{2}}{2}$ .

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Reference angle: $\sin 45° = \dfrac{\sqrt{2}}{2}$ , so reference angle $= 45°$ .

Sine is positive in the first and second quadrants.

Within $-180° \leq \theta \leq 180°$ :

\theta = 45° \quad \text{or} \quad \theta = 180° - 45° = 135°

Answer: $\theta = 45°$ or $\theta = 135°$ .

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Connections

Prerequisite — Basic right-angle trigonometry (SOHCAHTOA): The geometric derivations of exact values rely directly on reading ratios from labelled right-angled triangles.
Next — The sine and cosine rules: Problems combining these rules with non-right-angled triangles frequently require exact values at the substitution step.
Next — Trigonometric identities ( $\sin^2\theta + \cos^2\theta \equiv 1$ ): Verifying identities is far cleaner when exact values are substituted without rounding.
Next — Graphs of trigonometric functions: Recognising the key coordinates on $y = \sin\theta$ , $y = \cos\theta$ , $y = \tan\theta$ uses exact values at the standard angles.
Next — Solving trigonometric equations: Every equation whose solutions lie at $30°$ , $45°$ , $60°$ (or related angles) demands these exact values; CAST and symmetry arguments build directly on this topic.
Later — Integration of $\sin$ and $\cos$ (Pure 2): Definite integrals between these special angles yield exact answers only when the exact values are known.

Introduction

Core Concept

Deriving the values geometrically

Related angles and the CAST diagram

Key Formulae & Definitions

The fundamental exact values

Boundary (quadrantal) values for completeness

Worked Examples

Example 1 — Finding exact values of related angles

Example 2 — Evaluating a trigonometric expression exactly

Example 3 — Solving a trigonometric equation using exact values

Common Mistakes & Examiner Pitfalls

Practice Questions

Connections

Figures

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