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tags:
Cosecant Coterminal Angles Sine Trigonometry Trigonometry Series Unit Circle

Trig Functions with the Unit Circle Exercise 4

Author: Hannah Solomon

The following problem is solved in this video. It is recommended that you try to solve the problem before watching the video. You can click "Reveal Answer" to see the answer to the problem.

Problem: Find the exact value of \(\sin(-45^\circ)\) and \(\sec\left(-\dfrac{2\pi}{3}\right).\)

Answer: \(\sin(-45^\circ)=-\dfrac{\sqrt{2}}{2}\quad\) \(\sec\left(-\dfrac{2\pi}{3}\right)=-2\)

Solution Method: To find the trig value of an angle less than 0, we find its coterminal angle on the unit circle, that means between 0 and \(360^\circ\) (or 0 and \(2\pi\)). Since coterminal angles all have the same terminal side, the trig values of coterminal angles are the same. 
 
Coterminal Angle: Angles in standard position (with the initial side on the positive x-axis) that have a common terminal side. Coterminal angles differ by multiples of \(360^\circ\) or 2\(\pi\) in radians.
(For more info see the coterminal angles section.)

Since we're starting with a negative angle in degrees, we're going to add a full rotation, 360 degrees, until the angle is between 0 and \(360^\circ.\) \(-45^\circ+350^\circ=315^\circ.\)
This means that \(-45^\circ\) and \(315^\circ\) are coterminal angles and \(\sin(-45^circ = \sin(315^\circ).\) The coordinate at \(315^\circ\) is \(\left(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right).\) Sine is equal to the \(y\)-coordinate, so \(\sin(-45^\circ)=-\frac{\sqrt{2}}{2}.\)

For \(\sec\left(-\frac{2\pi}{3}\right),\) we want to find the coterminal angle to \(-\frac{2\pi}{3}\). So we add \(2\pi\) to \(\frac{2\pi}{3}\): 
    \begin{equation*}
        -\frac{2\pi}{3} +2\pi= -\frac{2\pi}{3}+\frac{6\pi}{3} = \frac{4\pi}{3}
    \end{equation*}
So \(-\frac{2\pi}{3}\) and \(\frac{4\pi}{3}\) are coterminal and \(\sec\left(-\frac{2\pi}{3}\right)=\sec\left(\frac{4\pi}{3}\right).\) The coordinate at \(\frac{4\pi}{3}\) is \(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right).\) Secant is the reciprocal of cosine and cosine is the x-coordinate, so we have
    \begin{equation*} 
        \sec\left(-\frac{2\pi}{3}\right) = \sec\left(\frac{4\pi}{3}\right) = \frac{1}{\cos{\frac{4\pi}{3}}} = \frac{1}{-\frac{1}{2}}=-2
    \end{equation*}
 

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