Question
find the angles in a right triangle with a hypotenuse of 9.2, and a leg of 7.7
Answers
GPT-4o mini
To find the angles in a right triangle with a hypotenuse of 9.2 and one leg measuring 7.7, we can use the trigonometric functions sine, cosine, and tangent.
1. **Find the first angle (let's call it \( A \)) using the cosine function:**
\[
\cos(A) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{7.7}{9.2}
\]
Calculating \( \cos(A) \):
\[
\cos(A) = \frac{7.7}{9.2} \approx 0.83695652
\]
Now, calculate angle \( A \):
\[
A = \cos^{-1}(0.83695652) \approx 33.20^\circ
\]
2. **Find the second angle (let's call it \( B \)) using the sine function:**
We can also find \( B \) directly using the triangle angle sum property, knowing that the angles in a right triangle sum to 90 degrees:
\[
B = 90^\circ - A \approx 90^\circ - 33.20^\circ \approx 56.80^\circ
\]
3. **Check with the tangent function for verification:**
We can find \( A \) using the sine function:
\[
\sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\text{opposite}}{9.2}
\]
We need to calculate the length of the opposite side (let's call it \( b \)) using the Pythagorean theorem:
\[
b^2 + 7.7^2 = 9.2^2
\]
Calculating \( 7.7^2 \) and \( 9.2^2 \):
\[
b^2 + 59.29 = 84.64 \implies b^2 = 25.35 \implies b \approx 5.03
\]
Now calculating \( \sin(A) \):
\[
\sin(A) = \frac{5.03}{9.2} \approx 0.546
\]
Calculating \( A \):
\[
A \approx \sin^{-1}(0.546) \approx 33.20^\circ
\]
So the angles in this right triangle are:
- \( A \approx 33.20^\circ \)
- \( B \approx 56.80^\circ \)
- \( C = 90^\circ \)
Therefore, the angles in the triangle are approximately:
- **Angle A:** \( 33.20^\circ \)
- **Angle B:** \( 56.80^\circ \)
- **Angle C:** \( 90^\circ \) (right angle)
1. **Find the first angle (let's call it \( A \)) using the cosine function:**
\[
\cos(A) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{7.7}{9.2}
\]
Calculating \( \cos(A) \):
\[
\cos(A) = \frac{7.7}{9.2} \approx 0.83695652
\]
Now, calculate angle \( A \):
\[
A = \cos^{-1}(0.83695652) \approx 33.20^\circ
\]
2. **Find the second angle (let's call it \( B \)) using the sine function:**
We can also find \( B \) directly using the triangle angle sum property, knowing that the angles in a right triangle sum to 90 degrees:
\[
B = 90^\circ - A \approx 90^\circ - 33.20^\circ \approx 56.80^\circ
\]
3. **Check with the tangent function for verification:**
We can find \( A \) using the sine function:
\[
\sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\text{opposite}}{9.2}
\]
We need to calculate the length of the opposite side (let's call it \( b \)) using the Pythagorean theorem:
\[
b^2 + 7.7^2 = 9.2^2
\]
Calculating \( 7.7^2 \) and \( 9.2^2 \):
\[
b^2 + 59.29 = 84.64 \implies b^2 = 25.35 \implies b \approx 5.03
\]
Now calculating \( \sin(A) \):
\[
\sin(A) = \frac{5.03}{9.2} \approx 0.546
\]
Calculating \( A \):
\[
A \approx \sin^{-1}(0.546) \approx 33.20^\circ
\]
So the angles in this right triangle are:
- \( A \approx 33.20^\circ \)
- \( B \approx 56.80^\circ \)
- \( C = 90^\circ \)
Therefore, the angles in the triangle are approximately:
- **Angle A:** \( 33.20^\circ \)
- **Angle B:** \( 56.80^\circ \)
- **Angle C:** \( 90^\circ \) (right angle)