What is the exact value of cos 195?

Steps to Solve

1. Identify the angle in terms of reference angle The angle 195 degrees is in the third quadrant. The reference angle can be found by subtracting 180 degrees from 195 degrees: $$ 195^\circ - 180^\circ = 15^\circ $$

2. Determine the cosine value in the third quadrant In the third quadrant, the cosine function is negative. Therefore, we use: $$ \cos(195^\circ) = -\cos(15^\circ) $$

3. Use the cosine of the reference angle To find $-\cos(15^\circ)$, we use the cosine angle subtraction identity: $$ \cos(15^\circ) = \cos(45^\circ - 30^\circ) $$ Using the identity: $$ \cos(a - b) = \cos a \cos b + \sin a \sin b $$ we find: $$ \cos(15^\circ) = \cos(45^\circ) \cos(30^\circ) + \sin(45^\circ) \sin(30^\circ $$

4. Substitute the known values Using known values: $$ \cos(45^\circ) = \frac{\sqrt{2}}{2}, , \cos(30^\circ) = \frac{\sqrt{3}}{2}, , \sin(45^\circ) = \frac{\sqrt{2}}{2}, , \sin(30^\circ) = \frac{1}{2} $$ Substituting these values in: $$ \cos(15^\circ) = \left(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2} \cdot \frac{1}{2}\right) $$ This simplifies to: $$ \cos(15^\circ) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4} $$

5. Find the value of cosine for 195 degrees Now we substitute back to find: $$ \cos(195^\circ) = -\cos(15^\circ) = -\frac{\sqrt{6} + \sqrt{2}}{4} $$