Question
In a right triangle, one angle is 63 degrees with a hypotenuse of 34 feet. What is the opposite of angle 63 in feet with 4 decimal places
Answers
GPT-4o mini
To find the length of the side opposite the 63-degree angle in a right triangle with a hypotenuse of 34 feet, we can use the sine function from trigonometry. The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.
The formula is:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
Where:
- \(\theta\) is the angle (63 degrees in this case)
- "opposite" is the length of the side opposite the angle
- "hypotenuse" is the length of the hypotenuse (34 feet)
We can rearrange this formula to solve for the opposite side:
\[
\text{opposite} = \sin(\theta) \times \text{hypotenuse}
\]
Substituting the known values:
\[
\text{opposite} = \sin(63^\circ) \times 34
\]
Using a calculator to find \(\sin(63^\circ)\):
\(\sin(63^\circ) \approx 0.8929\)
Now substituting this value into the equation:
\[
\text{opposite} \approx 0.8929 \times 34 \approx 30.3056
\]
Thus, the length of the side opposite to the 63-degree angle is approximately:
\[
\boxed{30.3056 \text{ feet}}
\]
The formula is:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
Where:
- \(\theta\) is the angle (63 degrees in this case)
- "opposite" is the length of the side opposite the angle
- "hypotenuse" is the length of the hypotenuse (34 feet)
We can rearrange this formula to solve for the opposite side:
\[
\text{opposite} = \sin(\theta) \times \text{hypotenuse}
\]
Substituting the known values:
\[
\text{opposite} = \sin(63^\circ) \times 34
\]
Using a calculator to find \(\sin(63^\circ)\):
\(\sin(63^\circ) \approx 0.8929\)
Now substituting this value into the equation:
\[
\text{opposite} \approx 0.8929 \times 34 \approx 30.3056
\]
Thus, the length of the side opposite to the 63-degree angle is approximately:
\[
\boxed{30.3056 \text{ feet}}
\]