Question
Solve for X in the right triangle
A right triangle shows three sides, one with a value of 36 cm and the other 77 cm, the third angle must be found.
A right triangle shows three sides, one with a value of 36 cm and the other 77 cm, the third angle must be found.
Answers
Answered by
GPT-4o mini
To solve for the angle \( X \) in a right triangle when you have two sides (one side measuring 36 cm and the other measuring 77 cm), we first need to determine which side corresponds to which.
Assuming that 36 cm and 77 cm are the lengths of the legs of the right triangle, we can use the tangent function to find the angle opposite the shorter leg (36 cm).
The tangent of angle \( X \) is given by the ratio of the length of the opposite side to the length of the adjacent side:
\[
\tan(X) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{36}{77}
\]
To find angle \( X \), we take the arctangent (inverse tangent) of the ratio:
\[
X = \tan^{-1}\left(\frac{36}{77}\right)
\]
Using a calculator to compute this:
\[
X \approx \tan^{-1}(0.4675) \approx 25.0° \text{ (rounded to one decimal place)}
\]
So, the angle \( X \) is approximately 25.0 degrees.
If instead, we consider 77 cm as the hypotenuse and 36 cm as one of the legs, we can find the angle opposite the 36 cm side using the sine function:
\[
\sin(X) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{36}{77}
\]
Then the angle would be found as follows:
\[
X = \sin^{-1}\left(\frac{36}{77}\right)
\]
Calculating this gives us a different angle.
Either way, ensure to specify which sides are the legs and which is the hypotenuse to clarify the situation. If you can define which side is the hypotenuse, we will need to apply either sine for finding the angle associated with the length of the hypotenuse or tangent appropriately.
Assuming that 36 cm and 77 cm are the lengths of the legs of the right triangle, we can use the tangent function to find the angle opposite the shorter leg (36 cm).
The tangent of angle \( X \) is given by the ratio of the length of the opposite side to the length of the adjacent side:
\[
\tan(X) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{36}{77}
\]
To find angle \( X \), we take the arctangent (inverse tangent) of the ratio:
\[
X = \tan^{-1}\left(\frac{36}{77}\right)
\]
Using a calculator to compute this:
\[
X \approx \tan^{-1}(0.4675) \approx 25.0° \text{ (rounded to one decimal place)}
\]
So, the angle \( X \) is approximately 25.0 degrees.
If instead, we consider 77 cm as the hypotenuse and 36 cm as one of the legs, we can find the angle opposite the 36 cm side using the sine function:
\[
\sin(X) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{36}{77}
\]
Then the angle would be found as follows:
\[
X = \sin^{-1}\left(\frac{36}{77}\right)
\]
Calculating this gives us a different angle.
Either way, ensure to specify which sides are the legs and which is the hypotenuse to clarify the situation. If you can define which side is the hypotenuse, we will need to apply either sine for finding the angle associated with the length of the hypotenuse or tangent appropriately.