To solve the expression [tex]\(4 \cos ^3 45^{\circ} - 3 \cos 45^{\circ} + \sin 45^{\circ}\)[/tex], let's proceed step-by-step.
1. Calculate [tex]\(\cos 45^\circ\)[/tex] and [tex]\(\sin 45^\circ\)[/tex]:
[tex]\[
\cos 45^{\circ} = \frac{1}{\sqrt{2}} = 0.7071067811865476
\][/tex]
[tex]\[
\sin 45^{\circ} = \frac{1}{\sqrt{2}} = 0.7071067811865475
\][/tex]
2. Calculate [tex]\(4 \cos^3 45^\circ\)[/tex]:
[tex]\(\cos 45^\circ\)[/tex] cubed is:
[tex]\[
\cos^3 45^{\circ} = (0.7071067811865476)^3 = 0.3535533905932737
\][/tex]
So, multiplying by 4:
[tex]\[
4 \cos^3 45^{\circ} = 4 \times 0.3535533905932737 = 1.4142135623730954
\][/tex]
3. Calculate [tex]\(3 \cos 45^\circ\)[/tex]:
[tex]\[
3 \cos 45^{\circ} = 3 \times 0.7071067811865476 = 2.121320343559643
\][/tex]
4. Sum all the terms:
Combine the terms:
[tex]\[
4 \cos^3 45^{\circ} - 3 \cos 45^{\circ} + \sin 45^{\circ} = 1.4142135623730954 - 2.121320343559643 + 0.7071067811865475
\][/tex]
5. Simplify the expression:
Adding and subtracting the terms:
[tex]\[
1.4142135623730954 - 2.121320343559643 + 0.7071067811865475 = 0.0
\][/tex]
Hence, the value of [tex]\(4 \cos ^3 45^{\circ} - 3 \cos 45^{\circ} + \sin 45^{\circ}\)[/tex] is [tex]\(0.0\)[/tex].
