Trigonometric functions and equations

Sine is zero at multiples of 180°. Find the corresponding x values and exclude both endpoints.

Graphical solutions of f(x) = a are the x coordinates where the graph meets y = a. Algebra often gives exact values while a graph generally gives approximations. Check every intersection and the specified interval.

The inner angle must change by one full sine period. Vertical scaling and shifting do not change the period.

Find α = arccos s. General solutions of cos x = s are x = ±α + 360°k, with integer k. Select all angles in [0°, 360°], including endpoints when valid.

Find α = arctan s. All solutions are x = α + 180°k, with integer k. Keep those in [0°, 360°]. For s = 0, include 0°, 180° and 360°; tangent is undefined at 90° and 270°.

Express the product using the double-angle sine. Apply the sine bounds, accounting for the coefficient and shift.

After half a sine period, sine changes sign but its square does not. Account for the coefficient of x; the phase shift does not affect the period.

Find α = arcsin s. General solutions of sin x = s are x = α + 360°k or x = 180° − α + 360°k, with integer k. Keep only angles in [0°, 360°] and remove duplicates.

Sine and cosine have period 360°; tangent has period 180°. sin(−α) = −sin α. Sine is positive in quadrants I and II and negative in III and IV; cosine is also negative in III. Special values: sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2.

The expression a sin x + b cos x can be written as R sin(x + φ), where R = √(a² + b²). Since the maximum value of sine is 1, the maximum value of the whole expression is √(a² + b²).