What is the exact value of cos(105)?

Steps to Solve

1. Express 105 degrees using known angles

To find $\cos(105^\circ)$, we can express it as the sum of two angles that we know the cosine values for. We can use the identity: $$ 105^\circ = 60^\circ + 45^\circ $$

2. Use the cosine angle addition formula

We will apply the cosine angle addition formula: $$ \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) $$ Here, let $a = 60^\circ$ and $b = 45^\circ$.

3. Substitute the known values

We know that:

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

Now substitute these values into the formula: $$ \cos(105^\circ) = \cos(60^\circ)\cos(45^\circ) - \sin(60^\circ)\sin(45^\circ) $$ This becomes: $$ \cos(105^\circ) = \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) - \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) $$

4. Simplify the expression

Now let's simplify the expression: $$ \cos(105^\circ) = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} $$ Combine the terms: $$ \cos(105^\circ) = \frac{\sqrt{2} - \sqrt{6}}{4} $$