Back to course
10.2PureCore

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 Trigonometric Identities and Equations
Edexcel Trig Identities & Equations — full chapter playlist (Zeeshan Zamurred)Curated by Kepler, not produced by us. Watch it before the notes below.
Open on YouTube

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
1

The two special triangles

A right-angled isosceles triangle with two sides of 11 gives the 45∘45^\circ values. Half of an equilateral triangle of side 22 gives the 30∘30^\circ and 60∘60^\circ values. From these, every exact value follows by Pythagoras.

sin⁡30∘=12,cos⁡30∘=32,tan⁡30∘=13\sin 30^\circ = \tfrac{1}{2}, \quad \cos 30^\circ = \tfrac{\sqrt{3}}{2}, \quad \tan 30^\circ = \tfrac{1}{\sqrt{3}}
From the half-equilateral triangle.
2

The values to memorise

These five angles cover almost every exact-value question. Notice the neat pattern in the sines: 02,12,22,32,42\tfrac{\sqrt{0}}{2}, \tfrac{\sqrt{1}}{2}, \tfrac{\sqrt{2}}{2}, \tfrac{\sqrt{3}}{2}, \tfrac{\sqrt{4}}{2} for 0∘0^\circ to 90∘90^\circ.

sin⁡45∘=cos⁡45∘=12,tan⁡45∘=1\sin 45^\circ = \cos 45^\circ = \tfrac{1}{\sqrt{2}}, \quad \tan 45^\circ = 1
sin⁡60∘=32\sin 60^\circ = \tfrac{\sqrt{3}}{2}, cos⁡60∘=12\cos 60^\circ = \tfrac{1}{2}, tan⁡60∘=3\tan 60^\circ = \sqrt{3}.
Example 1State the exact value of cos⁡30∘\cos 30^\circ.
1From the half-equilateral triangle.
Answer32\dfrac{\sqrt{3}}{2}

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

3

Combining with CAST

For angles outside 0∘0^\circ–90∘90^\circ, find the related acute angle, take its exact value, and attach the correct sign from the CAST diagram.

Example 1Find the exact value of cos⁡150∘\cos 150^\circ.
1Related acute angle =30∘= 30^\circ; 2nd quadrant ⟶ cos⁡\cos negative.
2cos⁡30∘=32\cos 30^\circ = \tfrac{\sqrt{3}}{2}.
Answer−32-\dfrac{\sqrt{3}}{2}

Formula recap

sin⁡30∘=12, cos⁡30∘=32\sin 30^\circ = \tfrac{1}{2}, \ \cos 30^\circ = \tfrac{\sqrt{3}}{2}
30° values.
sin⁡45∘=cos⁡45∘=12\sin 45^\circ = \cos 45^\circ = \tfrac{1}{\sqrt{2}}
45° values.
sin⁡60∘=32, cos⁡60∘=12\sin 60^\circ = \tfrac{\sqrt{3}}{2}, \ \cos 60^\circ = \tfrac{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.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

All 6 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.