Answer :
To determine the measure of an acute angle in a right triangle when you know the lengths of the opposite side and the adjacent side, you can use the tangent function from trigonometry.
The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. This relationship is expressed as:
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
Given:
- The length of the opposite side is [tex]\( 7 \)[/tex] inches.
- The length of the adjacent side is [tex]\( 4 \)[/tex] inches.
We need to find the angle [tex]\( \theta \)[/tex]. We start by calculating the ratio:
[tex]\[ \tan(\theta) = \frac{7}{4} \][/tex]
Next, we need to find the angle whose tangent is [tex]\( \frac{7}{4} \)[/tex]. This can be achieved by finding the inverse tangent (often denoted as [tex]\( \arctan \)[/tex] or [tex]\( \tan^{-1} \)[/tex]) of [tex]\( \frac{7}{4} \)[/tex]:
[tex]\[ \theta = \tan^{-1} \left( \frac{7}{4} \right) \][/tex]
This yields the angle in radians. However, we typically want the angle in degrees. Therefore, we convert the radians to degrees.
Performing this calculation, we get:
[tex]\[ \theta \approx 60.3^\circ \][/tex]
Thus, the measure of the angle is:
[tex]\[ \boxed{60.3^\circ} \][/tex]
So, the correct answer is:
60.3°.
The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. This relationship is expressed as:
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
Given:
- The length of the opposite side is [tex]\( 7 \)[/tex] inches.
- The length of the adjacent side is [tex]\( 4 \)[/tex] inches.
We need to find the angle [tex]\( \theta \)[/tex]. We start by calculating the ratio:
[tex]\[ \tan(\theta) = \frac{7}{4} \][/tex]
Next, we need to find the angle whose tangent is [tex]\( \frac{7}{4} \)[/tex]. This can be achieved by finding the inverse tangent (often denoted as [tex]\( \arctan \)[/tex] or [tex]\( \tan^{-1} \)[/tex]) of [tex]\( \frac{7}{4} \)[/tex]:
[tex]\[ \theta = \tan^{-1} \left( \frac{7}{4} \right) \][/tex]
This yields the angle in radians. However, we typically want the angle in degrees. Therefore, we convert the radians to degrees.
Performing this calculation, we get:
[tex]\[ \theta \approx 60.3^\circ \][/tex]
Thus, the measure of the angle is:
[tex]\[ \boxed{60.3^\circ} \][/tex]
So, the correct answer is:
60.3°.
