Questions: Use reference angles to find the exact value of the expression sin(-225°) sin(-225°) = (Type an exact answer, using radicals as needed.)

Transcript text: Use reference angles to find the exact value of the expression \[ \boldsymbol{\operatorname { s i n }}\left(-225^{\circ}\right) \] $\sin \left(-225^{\circ}\right)=$ $\square$ (Type an exact answer, using radicals as needed.)

Solution

Solution Steps

To find the exact value of $\sin(-225^\circ)$ using reference angles, follow these steps:

Determine the reference angle for $-225^\circ$.
Use the properties of the sine function and the unit circle to find the sine of the reference angle.
Adjust the sign based on the quadrant in which the angle lies.

Step 1: Determine the Positive Angle

To find $\sin(-225^\circ)$, we first convert the negative angle to a positive equivalent: \[ -225^\circ + 360^\circ = 135^\circ \]

Step 2: Find the Reference Angle

Next, we identify the reference angle for $135^\circ$. Since $135^\circ$ is in the second quadrant, the reference angle is: \[ 180^\circ - 135^\circ = 45^\circ \]

Step 3: Calculate the Sine Value

The sine of the reference angle $45^\circ$ is: \[ \sin(45^\circ) = \frac{\sqrt{2}}{2} \] Since $135^\circ$ is in the second quadrant, where sine is positive, we have: \[ \sin(135^\circ) = \frac{\sqrt{2}}{2} \]

Step 4: Adjust for the Original Angle

For the original angle $-225^\circ$, we find: \[ \sin(-225^\circ) = -\sin(135^\circ) = -\frac{\sqrt{2}}{2} \]