Trigonometric Ratios of Standard Angles
[MathDefinition title="Standard Angles"] Standard angles in trigonometry are 0°, 30°, 45°, 60°, and 90°. The trigonometric ratios for these angles have specific values that should be memorized. [/MathDefinition]
[MathDefinition title="Trigonometric Ratios"] For an acute angle θ in a right triangle:
- sin θ = Opposite/Hypotenuse
- cos θ = Adjacent/Hypotenuse
- tan θ = Opposite/Adjacent
- cosec θ = 1/sin θ
- sec θ = 1/cos θ
- cot θ = 1/tan θ [/MathDefinition]
[MathKeyFormula title="Trigonometric Values Table"]
| Angle | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan | 0 | 1/√3 | 1 | √3 | ∞ |
| [/MathKeyFormula] |
[MathNote title="Memory Trick"] For sin values, write 0, 1, 2, 3, 4 then divide each by 2 and take square root: √0/2, √1/2, √2/2, √3/2, √4/2 which gives 0, 1/2, 1/√2, √3/2, 1. [/MathNote]
[MathExample title="Example 1: Evaluating Expression"] Problem: Find the value of sin 60° cos 30° + cos 60° sin 30°.
Solution: We know that: sin 60° = √3/2, cos 30° = √3/2 cos 60° = 1/2, sin 30° = 1/2
sin 60° cos 30° + cos 60° sin 30° = (√3/2)(√3/2) + (1/2)(1/2) = 3/4 + 1/4 = 4/4 = 1
Therefore, sin 60° cos 30° + cos 60° sin 30° = 1 [/MathExample]
[MathExample title="Example 2: Simplifying Expression"] Problem: Evaluate: (sin² 45° + cos² 45°) / (tan² 60° − sin² 30°)
Solution: sin 45° = 1/√2, cos 45° = 1/√2 tan 60° = √3, sin 30° = 1/2
Numerator: sin² 45° + cos² 45° = (1/√2)² + (1/√2)² = 1/2 + 1/2 = 1
Denominator: tan² 60° − sin² 30° = (√3)² − (1/2)² = 3 − 1/4 = 11/4
Result = 1/(11/4) = 4/11 [/MathExample]
[MathExample title="Example 3: Finding Unknown Angle"] Problem: If sin A = 1/2 and A is an acute angle, find the values of cos A and tan A.
Solution: Given sin A = 1/2, we know A = 30°
cos A = cos 30° = √3/2 tan A = tan 30° = 1/√3
Alternatively, using the identity sin² A + cos² A = 1: cos² A = 1 − sin² A = 1 − (1/2)² = 1 − 1/4 = 3/4 cos A = √3/2 (since A is acute)
tan A = sin A/cos A = (1/2)/(√3/2) = 1/√3 [/MathExample]
[MathExample title="Example 4: Proving Identity"] Problem: Prove that tan 30° × tan 60° = 1.
Solution: We know that: tan 30° = 1/√3 tan 60° = √3
tan 30° × tan 60° = (1/√3) × √3 = √3/√3 = 1
Hence proved. [/MathExample]