Final answer:
To solve the trigonometric equations provided, one must utilize trigonometric identities and the periodic nature of trigonometric functions. The sine and cosine of complementary angles, as well as the periodicity of sine and cosine functions, are essential in determining the values of x that satisfy each equation.
Step-by-step explanation:
The student's question pertains to solving various trigonometric equations. To solve an equation like sin(x) = ±cos(x ∓90°), one must understand that trigonometric identities and functions' periodic nature are key. For instance, using the identity sin(x) = cos(90° - x), we can find solutions for x within the desired interval. Similarly, the equations cos(x) = ±sin(x ± 90°), sin(x) = -sin(x ± 180°), and cos(x) = -cos(x ± 180°) require using fundamental trigonometric identities such as cos(x) = sin(90° - x) and the fact that sine and cosine are periodic with a period of 360° to find solutions.
Key Trigonometric Identities
- Sine of sum or difference: sin(a ± β) = sin a cos β ± cos a sin β
- Cosine of sum or difference: cos(α ± β) = cos a cos β - sin a sin β
- Cosine and Sine of complementary angles: sin(x) = cos(90° - x) and cos(x) = sin(90° - x)
- sin(x) = -sin(x ± 180°) indicates a phase shift by half a period for the sine function.
- cos(x) = -cos(x ± 180°) indicates a phase shift by half a period for the cosine function.
Triangle Relations
- Law of Sines: relates the lengths of sides of a triangle to the sines of its angles.
- Law of Cosines: generalizes the Pythagorean theorem and relates the lengths of sides of a triangle to the cosine of one of its angles.