To solve for [tex]\( x \)[/tex] given that [tex]\(\tan(x) = \frac{6}{8}\)[/tex], follow these steps:
1. Calculate the Tangent Value:
First, simplify the fraction [tex]\(\frac{6}{8}\)[/tex].
[tex]\[
\frac{6}{8} = 0.75
\][/tex]
So, we have:
[tex]\[
\tan(x) = 0.75
\][/tex]
2. Use the Inverse Tangent Function:
To find the angle [tex]\( x \)[/tex], we use the inverse tangent (also known as arctangent). The inverse tangent function [tex]\(\tan^{-1}\)[/tex] or [tex]\(\arctan\)[/tex] will give us the angle whose tangent is 0.75.
[tex]\[
x = \tan^{-1}(0.75)
\][/tex]
3. Find the Angle in Radians:
The inverse tangent of 0.75 is approximately 0.6435011087932844 radians.
[tex]\[
x \approx 0.6435011087932844 \text{ radians}
\][/tex]
4. Convert the Angle from Radians to Degrees:
Since angle measurements are often more intuitive in degrees, we convert the angle from radians to degrees. Use the conversion factor [tex]\(180^\circ / \pi\)[/tex].
[tex]\[
x \approx 0.6435011087932844 \times \left(\frac{180}{\pi}\right)
\][/tex]
This simplifies to approximately 36.86989764584402 degrees.
[tex]\[
x \approx 36.86989764584402^\circ
\][/tex]
5. Round to the Nearest Hundredth:
Finally, round the angle [tex]\(x\)[/tex] to the nearest hundredth.
[tex]\[
x \approx 36.87^\circ
\][/tex]
So, the value of [tex]\( x \)[/tex] is approximately [tex]\( 36.87^\circ \)[/tex] when rounded to the nearest hundredth.
