Question
Solve for the missing side. Round to the nearest tenth. (1 point) Responses 3.9 3.9 5.7 5.7 4.5 4.5 27.0. X length, 24 degrees width, 11 angle. just do the equation 2 answers
Answers
GPT-4o mini
To solve for the missing side (X) in a right triangle, we can use the given angle (24 degrees) and the length of the side opposite to that angle (11 units). Given that this is a right triangle, we can apply the sine function for opposite side (X) and hypotenuse (11).
The sine function is defined as follows:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
Thus, for our triangle, we can rearrange the equation to solve for the opposite side (X):
\[
X = \text{hypotenuse} \cdot \sin(\theta)
\]
Substituting the values we have:
\[
X = 11 \cdot \sin(24^\circ)
\]
Calculating \( \sin(24^\circ) \) (using a calculator):
\[
\sin(24^\circ) \approx 0.4067
\]
Now plug that back into the equation for X:
\[
X \approx 11 \cdot 0.4067 \approx 4.4737
\]
Rounding to the nearest tenth, we get:
\[
X \approx 4.5
\]
Now, if you need another angle (for example, the adjacent side using cosine), we can calculate it as well:
Using the cosine function for adjacent side (A):
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
\[
A = \text{hypotenuse} \cdot \cos(\theta)
\]
Substituting values:
\[
A = 11 \cdot \cos(24^\circ)
\]
Calculating \( \cos(24^\circ) \):
\[
\cos(24^\circ) \approx 0.9063
\]
Now plug that back into the equation for A:
\[
A \approx 11 \cdot 0.9063 \approx 9.9693
\]
Rounding to the nearest tenth, we get:
\[
A \approx 10
\]
So the two answers for the missing side (considering the opposite side and adjacent) are:
1. \( X \approx 4.5 \)
2. \( A \approx 10 \)
The sine function is defined as follows:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
Thus, for our triangle, we can rearrange the equation to solve for the opposite side (X):
\[
X = \text{hypotenuse} \cdot \sin(\theta)
\]
Substituting the values we have:
\[
X = 11 \cdot \sin(24^\circ)
\]
Calculating \( \sin(24^\circ) \) (using a calculator):
\[
\sin(24^\circ) \approx 0.4067
\]
Now plug that back into the equation for X:
\[
X \approx 11 \cdot 0.4067 \approx 4.4737
\]
Rounding to the nearest tenth, we get:
\[
X \approx 4.5
\]
Now, if you need another angle (for example, the adjacent side using cosine), we can calculate it as well:
Using the cosine function for adjacent side (A):
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
\[
A = \text{hypotenuse} \cdot \cos(\theta)
\]
Substituting values:
\[
A = 11 \cdot \cos(24^\circ)
\]
Calculating \( \cos(24^\circ) \):
\[
\cos(24^\circ) \approx 0.9063
\]
Now plug that back into the equation for A:
\[
A \approx 11 \cdot 0.9063 \approx 9.9693
\]
Rounding to the nearest tenth, we get:
\[
A \approx 10
\]
So the two answers for the missing side (considering the opposite side and adjacent) are:
1. \( X \approx 4.5 \)
2. \( A \approx 10 \)
10 is not a option