Question

Find the exact value of tan Q in simplest radical form. the hypotenuse is 4 and the adjacent side is 2 the opposite side is the square root of 12

Answer 1

In a right triangle, the tangent of an angle \( Q \) is defined as the ratio of the length of the opposite side to the length of the adjacent side.

Given:
- Hypotenuse \( h = 4 \)
- Adjacent side \( a = 2 \)
- Opposite side \( o = \sqrt{12} \)

To find \( \tan Q \), we use the formula:

\[
\tan Q = \frac{\text{opposite}}{\text{adjacent}} = \frac{o}{a}
\]

Substituting the values:

\[
\tan Q = \frac{\sqrt{12}}{2}
\]

Next, we simplify \( \sqrt{12} \):

\[
\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}
\]

Now we substitute this back into the equation for \( \tan Q \):

\[
\tan Q = \frac{2\sqrt{3}}{2}
\]

The \( 2 \) in the numerator and denominator cancels out:

\[
\tan Q = \sqrt{3}
\]

Thus, the exact value of \( \tan Q \) in simplest radical form is

\[
\boxed{\sqrt{3}}
\]