Sure! Let's break down the solution step by step.
We are given the function [tex]\( y = 5 \cos (2x - 20) \)[/tex].
We need to evaluate this function at specific points. Let's choose the points [tex]\( x = 0 \)[/tex], [tex]\( x = \frac{\pi}{4} \)[/tex], and [tex]\( x = \frac{\pi}{2} \)[/tex].
1. Evaluate at [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 5 \cos(2 \cdot 0 - 20)
\][/tex]
Simplifying the argument of the cosine function:
[tex]\[
y = 5 \cos(-20)
\][/tex]
Evaluating [tex]\( \cos(-20) \)[/tex]:
[tex]\[
y \approx 2.0404103090669596
\][/tex]
2. Evaluate at [tex]\( x = \frac{\pi}{4} \)[/tex]:
[tex]\[
y = 5 \cos \left(2 \cdot \frac{\pi}{4} - 20 \right)
\][/tex]
Simplifying the argument of the cosine function:
[tex]\[
y = 5 \cos \left( \frac{\pi}{2} - 20 \right)
\][/tex]
Evaluating [tex]\( \cos \left( \frac{\pi}{2} - 20 \right) \)[/tex]:
[tex]\[
y \approx 4.564726253638135
\][/tex]
3. Evaluate at [tex]\( x = \frac{\pi}{2} \)[/tex]:
[tex]\[
y = 5 \cos \left( 2 \cdot \frac{\pi}{2} - 20 \right)
\][/tex]
Simplifying the argument of the cosine function:
[tex]\[
y = 5 \cos (\pi - 20)
\][/tex]
Evaluating [tex]\( \cos (\pi - 20) \)[/tex]:
[tex]\[
y \approx -2.040410309066959
\][/tex]
Putting it all together, the values of [tex]\( y \)[/tex] at the specified points are:
- At [tex]\( x = 0 \)[/tex], [tex]\( y \approx 2.0404103090669596 \)[/tex].
- At [tex]\( x = \frac{\pi}{4} \)[/tex], [tex]\( y \approx 4.564726253638135 \)[/tex].
- At [tex]\( x = \frac{\pi}{2} \)[/tex], [tex]\( y \approx -2.040410309066959 \)[/tex].
These are the evaluated results for the function [tex]\( y = 5 \cos (2x - 20) \)[/tex] at the given points.
