Question

use reference angles to find the exact value of the following expression. do not use a calculator.
(\tan 765^{circ})
determine the reference angle for (765^{circ})
the reference angle is (45^{circ}).
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. (\tan 765^{circ}=square)
(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression. rationalize the denominator)
b. the answer is undefined.

Explanation:

Step1: Reduce the angle by 360° multiples

Since the period of tangent function is \(360^\circ\), we subtract \(360^\circ\) twice from \(765^\circ\) (because \(2\times360^\circ = 720^\circ\)):
\(765^\circ - 720^\circ = 45^\circ\).
So, \(\tan 765^\circ=\tan(720^\circ + 45^\circ)=\tan 45^\circ\) (using the periodicity of tangent: \(\tan(\theta + 360^\circ n)=\tan\theta\) for integer \(n\)).

Step2: Evaluate \(\tan 45^\circ\)

We know that \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), and for \(\theta = 45^\circ\), \(\sin 45^\circ=\cos 45^\circ=\frac{\sqrt{2}}{2}\). Thus,
\(\tan 45^\circ=\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1\).

Answer:

A. \(\tan 765^\circ = \boldsymbol{1}\)