To find the exact value of [tex]\(\cos 75^\circ\)[/tex] without using a calculator, we can use the cosine addition formula. Specifically, for this angle, we can express it using the sum of two angles with well-known trigonometric values: [tex]\(45^\circ\)[/tex] and [tex]\(30^\circ\)[/tex]. The cosine addition formula is:
[tex]\[
\cos(a + b) = \cos(a) \cos(b) - \sin(a) \sin(b)
\][/tex]
For [tex]\(a = 45^\circ\)[/tex] and [tex]\(b = 30^\circ\)[/tex]:
[tex]\[
\cos 75^\circ = \cos(45^\circ + 30^\circ) = \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ
\][/tex]
First, let's recall the exact trigonometric values for these angles:
[tex]\[
\cos 45^\circ = \frac{\sqrt{2}}{2}, \quad \sin 45^\circ = \frac{\sqrt{2}}{2}
\][/tex]
[tex]\[
\cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \sin 30^\circ = \frac{1}{2}
\][/tex]
Now we can substitute these values into the cosine addition formula:
[tex]\[
\cos 75^\circ = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right)
\][/tex]
Simplify each term:
[tex]\[
\cos 75^\circ = \frac{\sqrt{2} \cdot \sqrt{3}}{4} - \frac{\sqrt{2} \cdot 1}{4}
\][/tex]
[tex]\[
\cos 75^\circ = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}
\][/tex]
Since we have a common denominator, we can combine the fractions:
[tex]\[
\cos 75^\circ = \frac{\sqrt{6} - \sqrt{2}}{4}
\][/tex]
Thus, the exact value of [tex]\(\cos 75^\circ\)[/tex] is:
[tex]\[
\cos 75^\circ = \frac{\sqrt{6} - \sqrt{2}}{4}
\][/tex]
So, our final form without using decimal approximations is:
[tex]\[
\cos 75^\circ = \frac{\sqrt{6} - \sqrt{2}}{4}
\][/tex]
