To find the value of [tex]\(\theta\)[/tex] given that [tex]\(\tan \theta = 24\)[/tex], we need to follow these steps:
1. Understand the Relationship:
The tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the adjacent side. Here, [tex]\(\tan \theta = 24\)[/tex].
2. Calculate the Angle in Radians:
To determine the angle [tex]\(\theta\)[/tex], we use the inverse tangent function ([tex]\(\arctan\)[/tex] or [tex]\(\tan^{-1}\)[/tex]). This function will provide the angle in radians. Let’s denote [tex]\(\theta\)[/tex] in radians as [tex]\(\theta_{\text{radians}}\)[/tex].
[tex]\[
\theta_{\text{radians}} = \arctan(24)
\][/tex]
3. Convert Radians to Degrees:
Angles in trigonometry are often more readable in degrees. To convert the radians to degrees, we use the fact that:
[tex]\[
1 \text{ radian} = \frac{180}{\pi} \text{ degrees}
\][/tex]
Thus, the angle in degrees is given by:
[tex]\[
\theta_{\text{degrees}} = \theta_{\text{radians}} \times \frac{180}{\pi}
\][/tex]
4. Performing the Calculation:
From the calculation steps, we know:
[tex]\[
\theta_{\text{radians}} \approx 1.5291537476963082
\][/tex]
Converting this to degrees:
[tex]\[
\theta_{\text{degrees}} \approx 87.61405596961119
\][/tex]
5. Round to One Decimal Place:
Finally, we round the angle to one decimal place:
[tex]\[
\theta_{\text{degrees}} \approx 87.6
\][/tex]
Therefore, the value of [tex]\(\theta\)[/tex] in degrees, rounded to one decimal place, is:
[tex]\[
\theta \approx 87.6^\circ
\][/tex]
