Section 1.3 : Trig Functions
3. Determine the exact value of \(\displaystyle \sin \left( {\frac{{7\pi }}{4}} \right)\) without using a calculator.
Show All Steps Hide All Steps
Hint : Sketch a unit circle and relate the angle to one of the standard angles in the first quadrant.
First we can notice that \(2\pi - \frac{\pi }{4} = \frac{{7\pi }}{4}\) and so the terminal line for \(\frac{{7\pi }}{4}\) will form an angle of \(\frac{\pi }{4}\) with the positive \(x\)-axis in the fourth quadrant and we’ll have the following unit circle for this problem.
Hint : Given the obvious symmetry in the unit circle relate the coordinates of the line representing \(\frac{{7\pi }}{4}\) to the coordinates of the line representing \(\frac{\pi }{4}\) and use those to answer the question.
The coordinates of the line representing \(\frac{{7\pi }}{4}\) will be the same as the coordinates of the line representing \(\frac{\pi }{4}\) except that the \(y\) coordinate will now be negative. So, our new coordinates will then be \(\left( {\frac{{\sqrt 2 }}{2}, - \frac{{\sqrt 2 }}{2}} \right)\) and so the answer is,
\[\sin \left( {\frac{{7\pi }}{4}} \right) = - \frac{{\sqrt 2 }}{2}\]