MathSetuThe trigonometric values of 0°, 30°, 45°, 60° and 90° are used in almost every question of Introduction to Trigonometry and Heights and Distances. You do not need to cram them: there is a pattern.
The sine pattern
Write the numbers 0, 1, 2, 3, 4 under the angles 0°, 30°, 45°, 60°, 90°. Take the square root of each, then divide by 2. That gives sin 0° = 0, sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2 and sin 90° = 1.
The cosine pattern
Cosine is the sine list written backwards: cos 0° = 1, cos 30° = √3/2, cos 45° = 1/√2, cos 60° = 1/2 and cos 90° = 0. This works because cos θ = sin (90° − θ).
The tangent
Do not learn tan separately. Use tan θ = sin θ / cos θ. For example, tan 30° = (1/2) ÷ (√3/2) = 1/√3, tan 45° = 1, and tan 60° = √3. Tan 90° is not defined, because cos 90° is zero.
The full table
- 0°: sin 0, cos 1, tan 0
- 30°: sin 1/2, cos √3/2, tan 1/√3
- 45°: sin 1/√2, cos 1/√2, tan 1
- 60°: sin √3/2, cos 1/2, tan √3
- 90°: sin 1, cos 0, tan not defined
Check yourself with the identity
Whenever you are unsure, test with sin²θ + cos²θ = 1. For 30°: (1/2)² + (√3/2)² = 1/4 + 3/4 = 1, so the pair is right.
How to practise
Cover the table and rebuild it from the pattern every day for a week. Then solve a few expressions such as sin 60° cos 30° + sin 30° cos 60°, which equals 1. The Introduction to Trigonometry chapter has solved examples, a formula bank and a chapter test, and the Applications of Trigonometry chapter uses these values to find heights and distances.