Question content area top left
Part 1
Solve the right triangle.
.
.
.
Question content area top right
Part 1
121 m
49.9 degrees
N
M
P
n
p
Question content area bottom
Part 1
Find the measure of M.
Mequals
  
40.1degrees ​(Round to one decimal place as​ needed.)
Part 2
The length of side nequals
  
enter your response here m and of side pequals
  
enter your response here m.
​(Round to the nearest integer as​ needed.)


To solve this right triangle problem, we can follow the given steps. Let's break it down:

### Step 1: Triangle Information
- The triangle \( \triangle NMP \) is a right triangle.
- We are given:
  - Angle \( \angle M = 40.1^\circ \)
  - Angle \( \angle N = 49.9^\circ \)
  - Hypotenuse \( NP = 121 \) m.

We need to find the lengths of sides \( n \) (adjacent to \( \angle N \)) and \( p \) (opposite \( \angle N \)).

### Step 2: Use Trigonometric Functions

#### Finding side \( n \) (adjacent to \( \angle N \)):
We can use the cosine function for this:
\[
\cos(49.9^\circ) = \frac{n}{121}
\]
Solving for \( n \):
\[
n = 121 \times \cos(49.9^\circ)
\]

#### Finding side \( p \) (opposite \( \angle N \)):
We can use the sine function for this:
\[
\sin(49.9^\circ) = \frac{p}{121}
\]
Solving for \( p \):
\[
p = 121 \times \sin(49.9^\circ)
\]

Now, let's calculate the values of \( n \) and \( p \).The calculations give us the following results:

- The length of side \( n \) (adjacent to \( \angle N \)) is approximately **78 meters**.
- The length of side \( p \) (opposite \( \angle N \)) is approximately **93 meters**.

Would you like more details on how these values were derived, or do you have any other questions?

Here are some related questions:
1. How can we verify these results using the Pythagorean theorem?
2. What trigonometric function is used to find the hypotenuse given two sides?
3. How do different angles affect the side lengths in a right triangle?
4. What are the other possible ways to solve a right triangle?
5. What is the significance of rounding in practical triangle problems?

**Tip**: When working with angles and trigonometric functions, always ensure your calculator is in the correct mode (degrees or radians).

Solve the right triangle given the angles and hypotenuse. Angle N = 49.9°, Angle M = 40.1°, Hypotenuse NP = 121 m. Find the lengths of sides n (adjacent to angle N) and p (opposite to angle N).

This problem involves solving a right triangle with given angles and hypotenuse. Using trigonometric functions (cosine and sine), learn how to find the lengths of the adjacent and opposite sides of the triangle. Suitable for students in Grades 9-12, this example includes step-by-step calculations and practical tips.

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