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Solve for the missing angles or values in the following trigonometric identities:

A.
[tex]\[
\begin{array}{l}
\sin \left(28^{\circ}\right) = \cos (\square) \\
\cos \left(33^{\circ}\right) = \sin (\square) \\
\cos \left(31^{\circ}\right) = \sin (\square) \\
\cos (90 - \theta) = \sin (\theta)
\end{array}
\][/tex]

Sagot :

GinnyAnsweridn
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]
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