Exercises: Symmetry and Periodicity of Trigonometric Functions

On the unit circle, the angle $x$ has terminal point $(a, b)$.
Which expression gives $\cos(x)$?

A function $f$ is called an even function if $f(-x) = f(x)$ for all $x$ in its domain. Which statement correctly describes an even function's graph?

A point travels counterclockwise around the unit circle.
After one complete revolution (adding $2\pi$ radians to the angle),
the terminal point returns to its starting position.

If $\sin\!\left(\dfrac{\pi}{6}\right) = \dfrac{1}{2}$,
what is $\sin\!\left(\dfrac{\pi}{6} + 2\pi\right)$?

Enter your answer as a fraction.

Use the even/odd properties of cosine and sine to evaluate.

Given that $\cos\!\left(\dfrac{\pi}{4}\right) = \dfrac{\sqrt{2}}{2}$,
find $\cos\!\left(-\dfrac{\pi}{4}\right)$.

Express your answer in simplest radical form.

Given that $\cos\!\left(\dfrac{2\pi}{3}\right) = -\dfrac{1}{2}$,
find $\cos\!\left(-\dfrac{2\pi}{3}\right)$.

Express your answer as a fraction.

Given that $\sin\!\left(\dfrac{5\pi}{6}\right) = \dfrac{1}{2}$,
find $\sin\!\left(-\dfrac{5\pi}{6}\right)$.

Express your answer as a fraction.

Which statement about the tangent function is correct?

Use the periodicity of cosine to evaluate.

Given that $\cos\!\left(\dfrac{\pi}{3}\right) = \dfrac{1}{2}$,
find $\cos\!\left(\dfrac{\pi}{3} + 6\pi\right)$.

Express your answer as a fraction.

The tangent function has period $\pi$.

Given that $\tan\!\left(\dfrac{\pi}{4}\right) = 1$,
which of the following equals $-1$?