Learn the trigonometric ratios of standard angles, and use the fundamental identities to evaluate expressions.
sin, cos, and tan have known exact values at 0°, 30°, 45°, 60°, and 90° — these are memorized and used throughout trigonometry.
Find the value of sin30° + cos60°.
1/2 + 1/2 = 1
Find the value of tan45° × tan60°.
1 × √3 = √3
Find the value of sin45° × cos45°.
(1/√2)×(1/√2)=1/2
sin²θ+cos²θ=1 is the fundamental identity, from which 1+tan²θ=sec²θ and 1+cot²θ=cosec²θ are derived.
Prove: (1-sinθ)(1+sinθ)=cos²θ
Expanding: 1-sin²θ=cos²θ, which is true by the fundamental identity
Simplify: sec²θ - tan²θ
By identity, this always equals 1
Simplify: (1+cotθ)² + (1−cotθ)²
=1+2cotθ+cot²θ+1−2cotθ+cot²θ=2+2cot²θ=2cosec²θ
1. Find the value of cos30° × cos60° - sin30° × sin60°.
2. Find the value of sin²45°+cos²45°.
3. Prove: sinθ/cosθ = tanθ using the ratio definitions.
4. Simplify: cosec²θ - cot²θ
5. Find the value of tan30° + tan60°.
6. Find the value of sin60° × cos30° + cos60° × sin30°.
7. Find the value of tan²45°−sin²30°.
8. Prove: sec²θ − 1 = tan²θ
9. Simplify: (1−cosθ)(1+cosθ)
10. Find the value of cos0° + sin90°.