To find the missing values in the trigonometric identities, let's use the property that relates the sine and cosine of complementary angles. This property states that [tex]\(\sin(\theta) = \cos(90^\circ - \theta)\)[/tex] and [tex]\(\cos(\theta) = \sin(90^\circ - \theta)\)[/tex].
1. First expression: [tex]\(\sin(28^\circ) = \cos(\square)\)[/tex]
To find the angle for this relationship, we use the property [tex]\(\sin(\theta) = \cos(90^\circ - \theta)\)[/tex]:
[tex]\[
\sin(28^\circ) = \cos(90^\circ - 28^\circ)
\][/tex]
Calculating [tex]\(90^\circ - 28^\circ\)[/tex]:
[tex]\[
90^\circ - 28^\circ = 62^\circ
\][/tex]
So, [tex]\(\sin(28^\circ) = \cos(62^\circ)\)[/tex].
Therefore, [tex]\(\square = 62^\circ\)[/tex].
2. Second expression: [tex]\(\cos(33^\circ) = \sin(\square)\)[/tex]
Here, we use the property [tex]\(\cos(\theta) = \sin(90^\circ - \theta)\)[/tex]:
[tex]\[
\cos(33^\circ) = \sin(90^\circ - 33^\circ)
\][/tex]
Calculating [tex]\(90^\circ - 33^\circ\)[/tex]:
[tex]\[
90^\circ - 33^\circ = 57^\circ
\][/tex]
So, [tex]\(\cos(33^\circ) = \sin(57^\circ)\)[/tex].
Therefore, [tex]\(\square = 57^\circ\)[/tex].
3. Third expression: [tex]\(\cos(31^\circ) = \sin(\square)\)[/tex]
Again, using the property [tex]\(\cos(\theta) = \sin(90^\circ - \theta)\)[/tex]:
[tex]\[
\cos(31^\circ) = \sin(90^\circ - 31^\circ)
\][/tex]
Calculating [tex]\(90^\circ - 31^\circ\)[/tex]:
[tex]\[
90^\circ - 31^\circ = 59^\circ
\][/tex]
So, [tex]\(\cos(31^\circ) = \sin(59^\circ)\)[/tex].
Therefore, [tex]\(\square = 59^\circ\)[/tex].
4. Fourth expression: [tex]\(\cos(90^\circ - \theta) = \sin(\theta)\)[/tex]
By definition, we know that this relationship is always true without additional computation.
To summarize, we have found the following identities:
- [tex]\(\sin(28^\circ) = \cos(62^\circ)\)[/tex]
- [tex]\(\cos(33^\circ) = \sin(57^\circ)\)[/tex]
- [tex]\(\cos(31^\circ) = \sin(59^\circ)\)[/tex]
- [tex]\(\cos(90^\circ - \theta) = \sin(\theta)\)[/tex]
Thus, the completed list of expressions is:
[tex]\[
\begin{array}{l}
\sin(28^\circ) = \cos(62^\circ) \\
\cos(33^\circ) = \sin(57^\circ) \\
\cos(31^\circ) = \sin(59^\circ) \\
\cos(90^\circ - \theta) = \sin(\theta)
\end{array}
\][/tex]
