To determine which trigonometric ratio to use when finding an unknown angle in a right triangle, given the lengths of the sides adjacent and opposite the angle, you need to understand the relationships between the sides and the trigonometric functions.
In a right triangle, the primary trigonometric ratios are defined as follows for an angle [tex]\( \theta \)[/tex]:
1. Sine (sin):
[tex]\[
\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}
\][/tex]
2. Cosine (cos):
[tex]\[
\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}
\][/tex]
3. Tangent (tan):
[tex]\[
\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}
\][/tex]
Given that you know the lengths of the sides adjacent and opposite to the angle, the appropriate ratio to use is tangent. This is because tangent directly relates the opposite side to the adjacent side.
The equation for the tangent of an angle [tex]\( \theta \)[/tex] is:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
To find the angle [tex]\( \theta \)[/tex], you would use the inverse tangent function:
[tex]\[
\theta = \tan^{-1} \left( \frac{\text{opposite}}{\text{adjacent}} \right)
\][/tex]
This inverse function is often referred to as arctan or atan.
Therefore, the correct trigonometric ratio to use is:
[tex]\[ \tan^{-1} \][/tex]
