HSF.TF.A.3 Tutor Intake — Special Triangles and Exact Trig Values

A 45-45-90 triangle has two legs of equal length. If both legs
have length 1, what is the length of the hypotenuse?

Which pair of values correctly gives $\sin\!\left(\frac{\pi}{4}\right)$
and $\cos\!\left(\frac{\pi}{4}\right)$?

A point at angle $x$ in the first quadrant of the unit circle has
coordinates $(a, b)$.

Write the coordinates of the point at angle $\pi - x$ and explain,
in one or two sentences, why the y-coordinate stays the same while
the x-coordinate changes sign.

Using the 30-60-90 triangle, what is the exact value of
$\sin\!\left(\frac{\pi}{6}\right)$?

Enter the value as a decimal or simplified fraction (e.g., enter
$0.5$ or $1/2$).

What is the exact value of $\tan\!\left(\frac{\pi}{6}\right)$?

Using the symmetry relation $\sin(\pi - x) = \sin(x)$, what is the
exact value of $\sin\!\left(\frac{5\pi}{6}\right)$?

What is the exact value of $\cos\!\left(\frac{7\pi}{6}\right)$?

(Hint: $\frac{7\pi}{6} = \pi + \frac{\pi}{6}$)

What is the exact value of $\cos\!\left(\frac{\pi}{3}\right)$?