To find the exact values of the given trigonometric expressions, we can use angle addition formulas for cosine.
1. Calculating \(\cos \left(\frac{5 \pi}{12}\right)\):
Let's break down \(\frac{5 \pi}{12}\) in terms of simpler known angles.
We know:
[tex]\[
\frac{5 \pi}{12} = \frac{\pi}{3} + \frac{\pi}{4}
\][/tex]
We will use the cosine addition formula:
[tex]\[
\cos(A + B) = \cos A \cos B - \sin A \sin B
\][/tex]
Here, \(A = \frac{\pi}{3}\) and \(B = \frac{\pi}{4}\).
We know the cosine and sine values for these angles:
[tex]\[
\cos \left(\frac{\pi}{3}\right) = \frac{1}{2}, \quad \sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}
\][/tex]
[tex]\[
\cos \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}, \quad \sin \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
\][/tex]
Substituting these values into the cosine addition formula:
[tex]\[
\cos \left(\frac{\pi}{3} + \frac{\pi}{4}\right) = \cos \left(\frac{\pi}{3}\right) \cos \left(\frac{\pi}{4}\right) - \sin \left(\frac{\pi}{3}\right) \sin \left(\frac{\pi}{4}\right)
\][/tex]
[tex]\[
= \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) - \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right)
\][/tex]
[tex]\[
= \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}
\][/tex]
[tex]\[
= -\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}
\][/tex]
Thus,
[tex]\[
\cos \left(\frac{5 \pi}{12}\right) = -\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}
\][/tex]
2. Calculating \(\cos \left(105^\circ\right)\):
We know that \(105^\circ\) can be expressed as \(60^\circ + 45^\circ\).
We use the same cosine addition formula:
[tex]\[
\cos (A + B) = \cos A \cos B - \sin A \sin B
\][/tex]
Here, \(A = 60^\circ\) and \(B = 45^\circ\).
We know the cosine and sine values for these angles:
[tex]\[
\cos (60^\circ) = \frac{1}{2}, \quad \sin (60^\circ) = \frac{\sqrt{3}}{2}
\][/tex]
[tex]\[
\cos (45^\circ) = \frac{\sqrt{2}}{2}, \quad \sin (45^\circ) = \frac{\sqrt{2}}{2}
\][/tex]
Substituting these values into the cosine addition formula:
[tex]\[
\cos (60^\circ + 45^\circ) = \cos (60^\circ) \cos (45^\circ) - \sin (60^\circ) \sin (45^\circ)
\][/tex]
[tex]\[
= \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) - \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right)
\][/tex]
[tex]\[
= \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}
\][/tex]
[tex]\[
= -\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}
\][/tex]
Thus,
[tex]\[
\cos (105^\circ) = -\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}
\][/tex]
In summary:
[tex]\[
\cos \left(\frac{5\pi}{12}\right) = -\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}
\][/tex]
[tex]\[
\cos (105^\circ) = -\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}
\][/tex]
