No, sin(180° - a) = sin(a), not sin(180°) = a × sin(a). This is the supplementary angle identity in trigonometry.
The correct trigonometric identity is: sin(180° - a) = sin(a)
This means:
- Sine of a supplementary angle equals the sine of the original angle
- Example: sin(150°) = sin(30°) = 0.5
- This reflects the symmetry of the sine function
Note that sin(180°) alone equals 0, not a multiplied by sin(a).
Importants Points:
- Correct identity: sin(180° - a) = sin(a)
- Reason: Supplementary angle property
- sin(180°) = 0 (a specific value)
Is cos(nπ) Equal to (-1)^n?
Yes, cos(nπ) = (-1)^n where n is any integer—this is a fundamental trigonometric identity.
This identity works because:
- When n is even: cos(nπ) = 1, and (-1)^n = 1
- When n is odd: cos(nπ) = -1, and (-1)^n = -1
Examples:
- n = 0: cos(0) = 1, (-1)^0 = 1 ✓
- n = 1: cos(π) = -1, (-1)^1 = -1 ✓
- n = 2: cos(2π) = 1, (-1)^2 = 1 ✓
- n = 3: cos(3π) = -1, (-1)^3 = -1 ✓
Importants Points:
- Identity: cos(nπ) = (-1)^n for integer n
- Pattern: Alternates between 1 and -1
- Applications: Fourier series, wave functions, calculus