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Multiple Angle Equation Calculator

Solve equations like sin(2θ) = 0.5 or cos(3θ) = -1. Find all solutions when the angle is multiplied.

Equation Setup

(θ )=

The Multiple Angle Effect

If you have f(nθ) = k, then nθ can take multiple values in [0°, 360n°), and dividing each by n gives θ in [0°, 360°).

Number of Solutions

For sin(nθ) or cos(nθ): typically 2n solutions in [0°, 360°)

For tan(nθ): typically n solutions in [0°, 360°)

Example: sin(2θ) = 0.5

2θ = 30°, 150° (in first period)

2θ = 30°, 150°, 390°, 510° (in [0°, 720°))

θ = 15°, 75°, 195°, 255°

When θ goes from 0° to 360°, 2θ goes from 0° to 720° - completing two full periods. This means the trig function hits each value twice as many times.

Use even/odd identities: sin(-nθ) = -sin(nθ) and cos(-nθ) = cos(nθ). Then solve with positive n and adjust signs.

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