Solve for [tex]\( x \)[/tex].

[tex]\[
\tan(x) = \frac{6}{8}
\][/tex]

[tex]\[ x = \tan^{-1} \left( \frac{6}{8} \right)^{\circ} \][/tex]

Hint: Use the inverse tangent function [tex]\(\left(\tan^{-1}\left(\frac{6}{8}\right)\right)\)[/tex].

Round to the nearest hundredth.



Answer :

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.