How should I interpret these trigonometry instructions?

Question

What is the difference between these two instructions? I can do the first one, but I am not sure how to do the second.

Without using a calculator, evaluate the following trigonometric functions for $\sin\theta$, $\cos\theta$, and $\tan\theta$.

For example: $$\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$$

By finding an angle coterminal to theta, find the cos, sin, and tan of the following.

for example: $$\cos\left(\frac{13\pi}{3}\right) = \frac{1}{2}$$

Answer 1

People keep making up words for simple math concepts all the time. But according to various sources,

Coterminal Angles are angles that share the same initial side and terminal side.

I would probably put this a bit differently, since an angle such as $\frac\pi3$ is not defined by exactly where on the plane you draw two sides of the angle. But if you are comparing two central angles constructed inside the unit circle such that the initial side is always the positive $x$ axis, then the angles will also have the same terminal side if the difference between the angles is an exact integer multiple of $2\pi.$

For example, if you turn $\frac\pi3$ radians, and then turn another $2\pi$ radians in addition to that, you will end up facing the same way as if you just turned $\frac\pi3$ radians and stopped.

So if you can add or subtract some multiple of $2\pi$ to or from the given angle to get an angle you know about, you have just used coterminal angles to help you answer the question.

In the case of $\frac{13\pi}{3}$, it is useful to subtract $4\pi,$ which is exactly two times $2\pi.$

Answer 2

We can determine those values using Reflections, shifts, and periodicity properties for trigonometric functions.

For example as noticed by Vasya in the comment, since $\cos \theta$ is periodic with period $2 \pi$ we have that $\cos \theta = \cos (2k\pi + \theta)$ and then

$$\cos\left(\frac{13\pi}{3}\right)=\cos\left(4\pi+\frac{\pi}{3}\right)=\cos\left(\frac{\pi}{3}\right)=\frac12$$

Answer 3

You can think of it this way. Consider a unit circle with the following divisions:

If we know that $\pi$ radians is just half a circle, then $\pi/3$ is just $1/3$ of half a circle; we can divide the top of the circle into three parts. The case for the bottom of the circle is similar.

Half a rotation is $\pi/3 + \pi/3 + \pi/3 = 3 \cdot \frac{\pi}{3} = \pi$ radians; one full rotation gives $\frac{6\pi}{3} = 2\pi$ radians. Full loops become a multiple of $6\pi$ in the numerator. (You can count backwards to consider negative angles such as $-\pi/3$.

You can continue counting after $6\pi/3$ to get coterminal angles to angles originally in the interval $0 \le \theta <2pi$. So $7\pi/3$ is in the same position as $\pi/3$ (and thus would have the same cosine ratio).

Hence we can count twice around so that we have $12\pi/3$ coterminal to $0$, and then count one more to get to $13\pi/3$ located in quadrant I with the same initial and terminal arm as $1\pi/3$.