Grasp Maths

Year 9

Exact trigonometric values

Know and use the exact values of sin⁡\sin, cos⁡\cos, and tan⁡\tan for 0°0°, 30°30°, 45°45°, 60°60°, and 90°90° without a calculator.

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Lesson overview

Geometry — trigonometry

The exact trigonometric values for special angles come from two key triangles: the **isosceles right-angled triangle** (for 45°45°) and the **equilateral triangle split in half** (for 30°30° and 60°60°). sin⁡30°=12,sin⁡45°=22,sin⁡60°=32\sin 30° = \dfrac{1}{2}, \quad \sin 45° = \dfrac{\sqrt{2}}{2}, \quad \sin 60° = \dfrac{\sqrt{3}}{2} cos⁡30°=32,cos⁡45°=22,cos⁡60°=12\cos 30° = \dfrac{\sqrt{3}}{2}, \quad \cos 45° = \dfrac{\sqrt{2}}{2}, \quad \cos 60° = \dfrac{1}{2} tan⁡30°=13,tan⁡45°=1,tan⁡60°=3\tan 30° = \dfrac{1}{\sqrt{3}}, \quad \tan 45° = 1, \quad \tan 60° = \sqrt{3} At 0°0° and 90°90°: sin⁡0°=0\sin 0° = 0, sin⁡90°=1\sin 90° = 1, cos⁡0°=1\cos 0° = 1, cos⁡90°=0\cos 90° = 0, tan⁡0°=0\tan 0° = 0, tan⁡90°\tan 90° is undefined.

The special triangles give us exact values

An equilateral triangle of side 22 split in half gives a right triangle with sides 11, 3\sqrt{3}, and 22.

$\sqrt{3}$$1$$2$

Half of an equilateral triangle of side 2

sin⁡30°\sin 30°12\dfrac{1}{2}

opposite/hypotenuse

cos⁡30°\cos 30°32\dfrac{\sqrt{3}}{2}

adjacent/hypotenuse

tan⁡30°\tan 30°13\dfrac{1}{\sqrt{3}}

opposite/adjacent

The ratios of sides in these special triangles give exact trig values.

The 45°45° isosceles right triangle

A square of side 11 cut along the diagonal gives a right triangle with two equal sides of 11 and hypotenuse 2\sqrt{2}.

$1$$1$$\sqrt{2}$$\sin 45° = \cos 45° = \dfrac{\sqrt{2}}{2}$

When the two shorter sides are equal, sin⁡=cos⁡\sin = \cos.

Memorising the pattern

There is a pattern: sin⁡\sin values go 02,12,22,32,42\dfrac{\sqrt{0}}{2}, \dfrac{\sqrt{1}}{2}, \dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{3}}{2}, \dfrac{\sqrt{4}}{2} for 0°,30°,45°,60°,90°0°, 30°, 45°, 60°, 90°.

$\theta$$0°$$30°$$45°$$60°$$90°$
$\sin \theta$$0$$\dfrac{1}{2}$$\dfrac{\sqrt{2}}{2}$$\dfrac{\sqrt{3}}{2}$$1$
$\cos \theta$$1$$\dfrac{\sqrt{3}}{2}$$\dfrac{\sqrt{2}}{2}$$\dfrac{1}{2}$$0$
$\tan \theta$$0$$\dfrac{1}{\sqrt{3}}$$1$$\sqrt{3}$undef.

cos⁡\cos values go in reverse order of sin⁡\sin. tan⁡=sin⁡cos⁡\tan = \dfrac{\sin}{\cos}.

Worked example

Without a calculator, find the exact value of 2sin⁡60°+cos⁡30°2\sin 60° + \cos 30°.

  1. sin⁡60°=32\sin 60° = \dfrac{\sqrt{3}}{2} and cos⁡30°=32\cos 30° = \dfrac{\sqrt{3}}{2}.
  2. 2sin⁡60°=2×32=32\sin 60° = 2 \times \dfrac{\sqrt{3}}{2} = \sqrt{3}.
  3. So the expression =3+32=23+32= \sqrt{3} + \dfrac{\sqrt{3}}{2} = \dfrac{2\sqrt{3} + \sqrt{3}}{2}.
  4. =332= \dfrac{3\sqrt{3}}{2}.

Try it

Recall the exact values from the table, then substitute carefully.

Question 1

What is the exact value of sin⁡30°\sin 30°?

Question 2

What is the exact value of cos⁡60°\cos 60°?

Question 3

What is the exact value of tan⁡45°\tan 45°?

Question 4

Without a calculator, find sin⁡230°+cos⁡230°\sin^2 30° + \cos^2 30°.

Question 5

Which trigonometric function is undefined at 90°90°?

Common mistakes

Watch for these when working through the lesson.

  • Confusing sin⁡30°=12\sin 30° = \frac{1}{2} with sin⁡60°=32\sin 60° = \frac{\sqrt{3}}{2}.
  • Forgetting that tan⁡90°\tan 90° is undefined (division by zero).
  • Writing 23\dfrac{\sqrt{2}}{3} instead of 32\dfrac{\sqrt{3}}{2} — the denominator is always 22 for sin⁡\sin and cos⁡\cos.

Related topics

These ideas fit closely with this lesson.

  • Pythagoras' theorem in 2D.
  • Similar shapes and scale factors.
  • Trigonometry in right-angled triangles.

Practice next

Independent practice will plug in here

This lesson builds the understanding first. Deeper adaptive practice can sit here later.