Without using a calculator, compute the sine and cosine of $225^{\circ}$ by using the reference angle.
What is the reference angle? degrees.
In what quadrant is this angle? (answer $1,2,3$, or 4 )
\[
\sin \left(225^{\circ}\right)=
\]
\[
\cos \left(225^{\circ}\right)=
\]
Answer
Therefore, \(\sin \left(225^\circ\right) = -\sin \left(45^\circ\right) = -\frac{\sqrt{2}}{2}\) and \(\cos \left(225^\circ\right) = -\cos \left(45^\circ\right) = -\frac{\sqrt{2}}{2}\).
Step 1 ：The reference angle is the acute angle that the given angle makes with the x-axis. For an angle of \(225^\circ\), the reference angle is \(225^\circ - 180^\circ = 45^\circ\).
Step 2 ：The angle \(225^\circ\) is in the third quadrant. In this quadrant, both sine and cosine are negative.
Step 3 ：Using the reference angle, we know that \(\sin(45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2}\).
Step 4 ：Therefore, \(\sin \left(225^\circ\right) = -\sin \left(45^\circ\right) = -\frac{\sqrt{2}}{2}\) and \(\cos \left(225^\circ\right) = -\cos \left(45^\circ\right) = -\frac{\sqrt{2}}{2}\).
