To solve for [tex]\(\theta\)[/tex] when [tex]\(\cos \theta = \frac{9}{40}\)[/tex], follow these steps:
1. Identify the cosine value:
[tex]\[
\cos \theta = \frac{9}{40} \approx 0.225
\][/tex]
2. Find [tex]\(\theta\)[/tex] in radians:
- To determine [tex]\(\theta\)[/tex], we need to apply the inverse cosine function (arccosine) to the cosine value:
[tex]\[
\theta = \arccos\left(\frac{9}{40}\right)
\][/tex]
- By using the arccos calculation, we obtain:
[tex]\[
\theta \approx 1.34385 \text{ radians}
\][/tex]
3. Convert radians to degrees:
- The conversion factor between radians and degrees is [tex]\(180^\circ / \pi\)[/tex]. To convert [tex]\(\theta\)[/tex] from radians to degrees, multiply by this factor:
[tex]\[
\theta_{\text{degrees}} = \theta \times \frac{180^\circ}{\pi}
\][/tex]
- Substituting [tex]\(\theta \approx 1.34385 \)[/tex] radians into the formula:
[tex]\[
\theta_{\text{degrees}} \approx 1.34385 \times \frac{180^\circ}{\pi} \approx 76.99712^\circ
\][/tex]
Thus, the solutions can be summarized as:
1. The cosine value: [tex]\( \cos \theta \approx 0.225 \)[/tex]
2. The angle [tex]\(\theta\)[/tex] in radians: [tex]\( \theta \approx 1.34385 \text{ radians} \)[/tex]
3. The angle [tex]\(\theta\)[/tex] in degrees: [tex]\( \theta_{\text{degrees}} \approx 76.99712^\circ \)[/tex]
Therefore, [tex]\(\theta \approx 1.34385 \text{ radians} \)[/tex] or [tex]\( \theta \approx 76.99712^\circ \)[/tex].
