To solve the right triangle ABC where angle C is the right angle, angle B is 75.0 degrees, and side [tex]\( b \)[/tex] (the side opposite angle B) is 129 inches, follow these steps:
1. Determine Angle A:
Since the sum of angles in any triangle is 180 degrees and angle C is 90 degrees in a right triangle:
[tex]\[
A = 90^\circ - B
\][/tex]
[tex]\[
A = 90^\circ - 75^\circ = 15^\circ
\][/tex]
2. Calculate Side [tex]\( a \)[/tex]:
To find the length of side [tex]\( a \)[/tex] (the side adjacent to angle B), use the trigonometric relationship involving the tangent function:
[tex]\[
\tan(B) = \frac{b}{a}
\][/tex]
Rearrange to solve for [tex]\( a \)[/tex]:
[tex]\[
a = \frac{b}{\tan(B)}
\][/tex]
Since tangent of 75 degrees is approximately 3.732, we have:
[tex]\[
a = \frac{129}{\tan(75^\circ)} \approx 34.6 \text{ inches}
\][/tex]
3. Calculate Side [tex]\( c \)[/tex]:
To find the hypotenuse [tex]\( c \)[/tex], use the Pythagorean theorem:
[tex]\[
c^2 = a^2 + b^2
\][/tex]
Substitute the known values:
[tex]\[
c^2 = 34.6^2 + 129^2
\][/tex]
Solve for [tex]\( c \)[/tex]:
[tex]\[
c \approx \sqrt{34.6^2 + 129^2} \approx 133.6 \text{ inches}
\][/tex]
Thus, the detailed results for the right triangle ABC are:
- Angle [tex]\( A \)[/tex] is [tex]\( 15^\circ \)[/tex]
- The length of side [tex]\( a \)[/tex] is approximately [tex]\( 34.6 \text{ inches} \)[/tex]
- The length of side [tex]\( c \)[/tex] is approximately [tex]\( 133.6 \text{ inches} \)[/tex]
