Cosine rule

Cosine rule

\cos(360-\theta)=\cos\theta

This means that, if we know one solution for \theta, we can find another by subtracting that angle from 360\degree.

Negative cosine angles

\cos(-\theta)=\cos\theta

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

Periodicity of cosine

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

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

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

… where n is any integer.

Solving cosine equations

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

• \cos^{-1}(0.5)=60\degree
• 360\degree-60\degree=300\degree
• 60\degree+360\degree=420\degree
• 300\degree+360\degree=660\degree
• Answer: \theta=60\degree,300\degree,420\degree,660\degree

Solve cos\theta=-0.5 for values of \theta between -360\degree and 360\degree.

• \cos^{-1}(-0.5)=120\degree
• 360\degree-120\degree=240\degree
• 120\degree-360\degree=-240\degree
• 240\degree-360\degree=-120\degree
• Answer: \theta=-240\degree,-120\degree,120\degree,240\degree

Solve cos\theta=\frac{\sqrt{2}}{2} for values of \theta between 0\degree and 360\degree.

• \cos^{-1}(\frac{\sqrt{2}}{2})=45\degree
• 360\degree-45\degree=315\degree
• Answer: \theta=45\degree,315\degree