To determine the values of the trigonometric functions given in the question, let's go through a step-by-step explanation.
1. Calculate [tex]\(\cos(420^\circ)\)[/tex]:
- First, notice that 420 degrees is more than 360 degrees. To bring it within the standard range of 0 to 360 degrees, subtract 360 degrees from 420 degrees:
[tex]\[
420^\circ - 360^\circ = 60^\circ
\][/tex]
- So, [tex]\(\cos(420^\circ) = \cos(60^\circ)\)[/tex].
- The value of [tex]\(\cos(60^\circ)\)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
2. Calculate [tex]\(\sin(450^\circ)\)[/tex]:
- Similarly, 450 degrees is more than 360 degrees. To bring it within the standard range, subtract 360 degrees from 450 degrees:
[tex]\[
450^\circ - 360^\circ = 90^\circ
\][/tex]
- So, [tex]\(\sin(450^\circ) = \sin(90^\circ)\)[/tex].
- The value of [tex]\(\sin(90^\circ)\)[/tex] is [tex]\(1\)[/tex].
Given these calculations, we can fill in the drop-down menus as follows:
- For [tex]\(\cos(420^\circ)\)[/tex], the correct option is [tex]\(\frac{1}{2}\)[/tex].
- For [tex]\(\sin(450^\circ)\)[/tex], the correct option is [tex]\(1\)[/tex].
Thus, the complete and correct options are:
[tex]\[
\begin{array}{l}
\cos \left(420^{\circ}\right)=1 / 2 \vee \\
\sin \left(450^{\circ}\right)=\checkmark \\
1 / 2 \\
-1 \\
-1 / 2 \\
1 \\
\end{array}
\][/tex]
