Let's fill in the table step by step with the correct values for each column: [tex]\( \cos \theta \)[/tex] and [tex]\( \sin (90^\circ - \theta) \)[/tex].
### Step-by-Step Computation:
1. For [tex]\( \theta = 0^\circ \)[/tex]:
- [tex]\( \cos 0^\circ = 1.0 \)[/tex]
- [tex]\( 90^\circ - 0^\circ = 90^\circ \)[/tex]
- [tex]\( \sin 90^\circ = 1.0 \)[/tex]
2. For [tex]\( \theta = 30^\circ \)[/tex]:
- [tex]\( \cos 30^\circ \approx 0.866 \)[/tex]
- [tex]\( 90^\circ - 30^\circ = 60^\circ \)[/tex]
- [tex]\( \sin 60^\circ \approx 0.866 \)[/tex]
3. For [tex]\( \theta = 60^\circ \)[/tex]:
- [tex]\( \cos 60^\circ \approx 0.500 \)[/tex]
- [tex]\( 90^\circ - 60^\circ = 30^\circ \)[/tex]
- [tex]\( \sin 30^\circ \approx 0.500 \)[/tex]
4. For [tex]\( \theta = 90^\circ \)[/tex]:
- [tex]\( \cos 90^\circ \approx 0.0 \)[/tex]
- [tex]\( 90^\circ - 90^\circ = 0^\circ \)[/tex]
- [tex]\( \sin 0^\circ = 0.0 \)[/tex]
5. For [tex]\( \theta = 120^\circ \)[/tex]:
- [tex]\( \cos 120^\circ \approx -0.500 \)[/tex]
- [tex]\( 90^\circ - 120^\circ = -30^\circ \)[/tex]
- [tex]\( \sin -30^\circ \approx -0.500 \)[/tex]
6. For [tex]\( \theta = 150^\circ \)[/tex]:
- [tex]\( \cos 150^\circ \approx -0.866 \)[/tex]
- [tex]\( 90^\circ - 150^\circ = -60^\circ \)[/tex]
- [tex]\( \sin -60^\circ \approx -0.866 \)[/tex]
7. For [tex]\( \theta = 180^\circ \)[/tex]:
- [tex]\( \cos 180^\circ = -1.0 \)[/tex]
- [tex]\( 90^\circ - 180^\circ = -90^\circ \)[/tex]
- [tex]\( \sin -90^\circ = -1.0 \)[/tex]
### Completed Table:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
\theta & \cos \theta & 90^\circ - \theta & \sin (90^\circ - \theta) \\
\hline
0^\circ & 1.0 & 90^\circ & 1.0 \\
\hline
30^\circ & 0.866 & 60^\circ & 0.866 \\
\hline
60^\circ & 0.500 & 30^\circ & 0.500 \\
\hline
90^\circ & 0.0 & 0^\circ & 0.0 \\
\hline
120^\circ & -0.500 & -30^\circ & -0.500 \\
\hline
150^\circ & -0.866 & -60^\circ & -0.866 \\
\hline
180^\circ & -1.0 & -90^\circ & -1.0 \\
\hline
\end{array}
\][/tex]
