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Exact Values of Trigonometric Ratios

For a handful of special angles — 0°, 30°, 45°, 60° and 90° — the trig ratios have exact surd values you should know by heart. They come from two simple triangles and let you give exact (non-decimal) answers.

25 min Video by Zeeshan Zamurred Trigonometric Identities and Equations

Edexcel Trig Identities & Equations — full chapter playlist (Zeeshan Zamurred)Watch the full walkthrough before the notes below.

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What you'll be able to do

Recall the exact values for 0°, 30°, 45°, 60°, 90°
Derive them from the special triangles
Combine exact values with the CAST diagram
Give exact answers in surd form

The two special triangles

A right-angled isosceles triangle with two sides of $1$ gives the $4 5^{\circ}$ values. Half of an equilateral triangle of side $2$ gives the $3 0^{\circ}$ and $6 0^{\circ}$ values. From these, every exact value follows by Pythagoras.

sin 3 0^{\circ} = \frac{1}{2}, cos 3 0^{\circ} = \frac{3}{2}, tan 3 0^{\circ} = \frac{1}{3}

From the half-equilateral triangle.

The values to memorise

These five angles cover almost every exact-value question. Notice the neat pattern in the sines: $\frac{0}{2}, \frac{1}{2}, \frac{2}{2}, \frac{3}{2}, \frac{4}{2}$ for $0^{\circ}$ to $9 0^{\circ}$ .

sin 4 5^{\circ} = cos 4 5^{\circ} = \frac{1}{2}, tan 4 5^{\circ} = 1

sin 6 0^{\circ} = \frac{3}{2}

cos 6 0^{\circ} = \frac{1}{2}

tan 6 0^{\circ} = 3

1From the half-equilateral triangle.

Answer

\frac{3}{2}

Tip — tan = sin ÷ cos for each angle, so you can rebuild any tan value from the sin and cos.

Combining with CAST

For angles outside $0^{\circ}$ – $9 0^{\circ}$ , find the related acute angle, take its exact value, and attach the correct sign from the CAST diagram.

1Related acute angle

= 3 0^{\circ}

; 2nd quadrant ⟶

cos

negative.

cos 3 0^{\circ} = \frac{3}{2}

Answer

- \frac{3}{2}

Formula recap

sin 3 0^{\circ} = \frac{1}{2}, cos 3 0^{\circ} = \frac{3}{2}

30° values.

sin 4 5^{\circ} = cos 4 5^{\circ} = \frac{1}{2}

45° values.

sin 6 0^{\circ} = \frac{3}{2}, cos 6 0^{\circ} = \frac{1}{2}

60° values.

Common mistakes to avoid

Mixing up sin 30° and cos 30°.

sin 30° = ½ (small), cos 30° = √3/2 (large). Sine grows as the angle grows.

Giving a rounded decimal when an exact value is asked for.

Leave it in surd form, e.g. √3/2, not 0.866.

Key takeaways

Know the exact values for 0°, 30°, 45°, 60°, 90°.
They come from the 45° (isosceles) and 30–60° (half-equilateral) triangles.
tan = sin/cos for each special angle.
For other angles, use the related acute angle + CAST sign.

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