Question

Solve for X, Round to the nearest tenth if necessary

triangle NOP
Hypotenuse 83
Angle 41

Answer 1

In triangle NOP, we are given the hypotenuse (c) as 83 and one of the angles (A) as 41 degrees. We can use trigonometric ratios to find the other sides of the triangle, specifically the opposite side (a) and the adjacent side (b) to angle A.

1. **Finding the opposite side (a):**
\[
\sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} \Rightarrow \sin(41^\circ) = \frac{a}{83}
\]
\[
a = 83 \cdot \sin(41^\circ)
\]
Now, let's calculate this value.

2. **Finding the adjacent side (b):**
\[
\cos(A) = \frac{\text{adjacent}}{\text{hypotenuse}} \Rightarrow \cos(41^\circ) = \frac{b}{83}
\]
\[
b = 83 \cdot \cos(41^\circ)
\]

Now we can calculate the values for \( a \) and \( b \):

Using a calculator:
- \(\sin(41^\circ) \approx 0.6561\)
- \(\cos(41^\circ) \approx 0.7547\)

Calculating \( a \):
\[
a \approx 83 \cdot 0.6561 \approx 54.4
\]

Calculating \( b \):
\[
b \approx 83 \cdot 0.7547 \approx 62.7
\]

So, rounding to the nearest tenth:
- The length of the side opposite to the 41-degree angle is approximately **54.4**.
- The length of the adjacent side is approximately **62.7**.

Would you like any more information on this triangle?