Finding all cosine angles

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

flashcards

Question	Answer
What is the cosine rule for finding angles related to 360-\theta?	\cos(360\degree-\theta) = \cos\theta
What is the rule for the cosine of a negative angle?	\cos(-\theta) = \cos\theta
What is the period of the cosine function?	360\degree
How do you find more solutions to cosine equations using periodicity?	Add or subtract 360\degree any number of times: \cos\theta = \cos(\theta + 360n) where n is any integer.
Solve \cos\theta = 0.5 for 0\degree \leq \theta \leq 720\degree.	List the four solutions. 60\degree, 300\degree, 420\degree, 660\degree
To solve \cos\theta = 0.5, what is the first step and result?	\cos^{-1}(0.5) = 60\degree
To solve \cos\theta = 0.5, how do you find the second solution in the first cycle?	360\degree - 60\degree = 300\degree
To solve \cos\theta = 0.5, how do you find solutions in the next period?	Add 360\degree to each of the first two solutions, giving 60\degree+360\degree=420\degree and 300\degree+360\degree=660\degree.
Solve \cos\theta = -0.5 for -360\degree \leq \theta \leq 360\degree.	List the four solutions. -240\degree, -120\degree, 120\degree, 240\degree
To solve \cos\theta = -0.5, what is the principal angle and its associated angle in the first positive cycle?	Principal: \cos^{-1}(-0.5) = 120\degree. Associated: 360\degree - 120\degree = 240\degree.
To solve \cos\theta = -0.5, how do you find the two negative solutions?	Subtract 360\degree from each positive solution: 120\degree - 360\degree = -240\degree and 240\degree - 360\degree = -120\degree.
Solve \cos\theta = \frac{\sqrt{2}}{2} for 0\degree \leq \theta \leq 360\degree.	List the two solutions. 45\degree, 315\degree
To solve \cos\theta = \frac{\sqrt{2}}{2}, what is the principal angle?	\cos^{-1}(\frac{\sqrt{2}}{2}) = 45\degree
To solve \cos\theta = \frac{\sqrt{2}}{2}, how do you find the second solution in the range 0\degree to 360\degree?	360\degree - 45\degree = 315\degree