What is the exact value of cos 105 degrees?

Steps to Solve

1. Identify the cosine angle We need to find the exact value of $\cos(105^\circ)$. We can break this angle down into smaller angles that we know the cosine values for.

2. Use the cosine angle addition formula We can express $105^\circ$ as $60^\circ + 45^\circ$. Thus, we can use the cosine addition formula: $$ \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) $$ In this case, $a = 60^\circ$ and $b = 45^\circ$.

3. Calculate cosine and sine values We need the cosine and sine values of $60^\circ$ and $45^\circ$:

• $\cos(60^\circ) = \frac{1}{2}$
• $\sin(60^\circ) = \frac{\sqrt{3}}{2}$
• $\cos(45^\circ) = \frac{\sqrt{2}}{2}$
• $\sin(45^\circ) = \frac{\sqrt{2}}{2}$

4. Substitute values into the cosine formula Now plug the values into the cosine addition formula: $$ \cos(105^\circ) = \cos(60^\circ)\cos(45^\circ) - \sin(60^\circ)\sin(45^\circ) $$ Substituting the known values: $$ \cos(105^\circ) = \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) - \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) $$

5. Simplify the expression Simplify the expression: $$ = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} $$ Combine the terms: $$ = \frac{\sqrt{2} - \sqrt{6}}{4} $$