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 \\( \square ^ { \circ } \\).

Explanation:

Step1: Subtract multiples of 360°

To find the coterminal angle of \(765^\circ\) within one full rotation (\(0^\circ\) to \(360^\circ\)), we subtract multiples of \(360^\circ\). Let's find how many times \(360^\circ\) fits into \(765^\circ\).
\(765\div360 = 2\) with a remainder. So we subtract \(2\times360^\circ = 720^\circ\) from \(765^\circ\).
\(765^\circ - 720^\circ = 45^\circ\)
Wait, no, wait. Wait, \(2\times360 = 720\), \(765 - 720 = 45\)? But wait, maybe I made a mistake. Wait, \(360\times2 = 720\), \(765 - 720 = 45\). But wait, is that correct? Wait, no, wait, \(360\times2 = 720\), \(765 - 720 = 45\). But wait, let's check again. Wait, the period of tangent is \(180^\circ\), but the reference angle is found by subtracting multiples of \(360^\circ\) to get an angle between \(0^\circ\) and \(360^\circ\), then finding the reference angle. Wait, \(765^\circ - 2\times360^\circ = 765 - 720 = 45^\circ\)? Wait, no, \(360\times2 = 720\), \(765 - 720 = 45\). But \(45^\circ\) is already in the first quadrant, so its reference angle is itself. Wait, but maybe I made a mistake. Wait, let's do it again. The formula for finding the coterminal angle is \(\theta - 360^\circ\times n\), where \(n\) is an integer, such that the result is between \(0^\circ\) and \(360^\circ\). So for \(765^\circ\), let's find \(n\) such that \(765 - 360n\) is between \(0\) and \(360\). Let's try \(n = 2\): \(765 - 720 = 45\), which is between \(0\) and \(360\). So the coterminal angle is \(45^\circ\), which is in the first quadrant, so the reference angle is \(45^\circ\). Wait, but let's check again. Wait, \(765^\circ\) is equal to \(2\times360^\circ + 45^\circ\), so the terminal side is the same as \(45^\circ\), so the reference angle is \(45^\circ\).

Wait, but maybe I made a mistake. Let's check with another approach. The reference angle is the acute angle formed by the terminal side of the given angle and the x - axis. For an angle \(\theta\) in standard position, if \(\theta\) is in the first quadrant (\(0^\circ<\theta<90^\circ\)), the reference angle is \(\theta\); if in the second quadrant (\(90^\circ<\theta<180^\circ\)), reference angle is \(180^\circ - \theta\); third quadrant (\(180^\circ<\theta<270^\circ\)), reference angle is \(\theta - 180^\circ\); fourth quadrant (\(270^\circ<\theta<360^\circ\)), reference angle is \(360^\circ - \theta\).

First, find the coterminal angle of \(765^\circ\) between \(0^\circ\) and \(360^\circ\). As we did before, \(765 - 2\times360 = 45^\circ\), which is in the first quadrant, so the reference angle is \(45^\circ\).

Wait, but let's confirm. Let's take \(765^\circ\), subtract \(360^\circ\) once: \(765 - 360 = 405^\circ\). Subtract \(360^\circ\) again: \(405 - 360 = 45^\circ\). So yes, the coterminal angle is \(45^\circ\), which is in the first quadrant, so the reference angle is \(45^\circ\).

Step2: Confirm the reference angle

Since the coterminal angle of \(765^\circ\) is \(45^\circ\) (by subtracting \(2\) multiples of \(360^\circ\)), and \(45^\circ\) lies in the first quadrant where the reference angle of an angle \(\alpha\) (where \(0^\circ<\alpha<90^\circ\)) is \(\alpha\) itself, the reference angle for \(765^\circ\) is \(45^\circ\).

Answer:

The reference angle is \(\boxed{45}\)°.