Finding all sine angles

Sine

\sin(180-\theta)=\sin\theta

This means that, if we know one solution for \theta, we can find another by subtracting that angle from 180\degree.

Negative sine angles

\sin(-\theta)=-\sin\theta

The sine of a negative angle is the negative of the sine of the positive angle

Periodicity of sine

The sine function has a period of 360\degree. This means that every interval of 360\degree, the sine function repeats its values.

This means that we can add or subtract 360\degree any number of times to find more solutions to sine equations:

\sin\theta=\sin(\theta+360n)

Solving sine equations

Solve sin\theta=0.5 for values of \theta between 0\degree and 360\degree.

• \sin^{-1}(0.5)=30\degree
• 180\degree-30\degree=150\degree
• Answer: \theta=30\degree,150\degree

Solve sin\theta=0.5 for values of \theta between 0\degree and 720\degree.

• \sin^{-1}(0.5)=30\degree
• 180\degree-30\degree=150\degree
• 30\degree+360\degree=390\degree
• 150\degree+360\degree=510\degree
• Answer: \theta=30\degree,150\degree,390\degree,510\degree

Solve sin\theta=-0.5 for values of \theta between 0\degree and 720\degree.

• \sin^{-1}(-0.5)=-30\degree
• 180\degree-(-30\degree)=210\degree
• -30\degree+360\degree=330\degree
• 210\degree+360\degree=570\degree
• 330\degree+360\degree=690\degree
• Answer: \theta=210\degree,330\degree,570\degree,690\degree

Solve sin\theta=\frac{\sqrt{3}}{2} for values of \theta between -360\degree and 360\degree.

• \sin^{-1}(\frac{\sqrt{3}}{2})=60\degree
• 180\degree-60\degree=120\degree
• 60\degree-360\degree=-300\degree
• 120\degree-360\degree=-240\degree
• Answer: \theta=-300\degree,-240\degree,60\degree,120\degree

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