Problem 61 - Precalculus Essentials

Question

In Exercises \(61-86,\) use reference angles to find the exact value of each expression. Do not use a calculator. $$\cos 225^{\circ}$$

Step-by-Step Solution

Answer

The exact value of \( \cos(225^{\circ}) \) is \( -\frac{\sqrt{2}}{2} \)

Step 1: Identifying the Quadrant

225 degrees is more than 180 degrees but less than 270 degrees, so it lies in the third quadrant of the unit circle.

Step 2: Finding the Reference Angle

The reference angle is found by subtracting the nearest multiple of 180 degrees from the given angle. So, the reference angle is: \(225^{\circ} - 180^{\circ} = 45^{\circ}\).

Step 3: Finding the cos value from the Reference Angle

In the unit circle, \( \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \)

Step 4: Assign the correct sign

In the third quadrant, both cosine and sine values are negative. Therefore, the final answer will be negative.

Step 5: Put all together

So, the exact value of \( \cos(225^{\circ}) \) is \( -\frac{\sqrt{2}}{2} \)