To solve for [tex]\( n \)[/tex] using the Law of Cosines, let's follow these steps:
Given:
- [tex]\( b = 22 \)[/tex] (one side of the triangle)
- [tex]\( c = 29 \)[/tex] (another side of the triangle)
- [tex]\( A = 41^\circ \)[/tex] (the angle between sides [tex]\( b \)[/tex] and [tex]\( c \)[/tex])
We need to find the length of side [tex]\( a \)[/tex] to the nearest whole number.
### Step 1: Convert the angle to radians
Since the cosine function in trigonometry is typically used with radians, we need to convert [tex]\( 41^\circ \)[/tex] to radians.
[tex]\[ A = 41^\circ \][/tex]
[tex]\[ A \text{ in radians} = 41 \times \frac{\pi}{180} \][/tex]
### Step 2: Apply the Law of Cosines
The Law of Cosines formula is:
[tex]\[ a^2 = b^2 + c^2 - 2bc \cos(A) \][/tex]
Substitute the given values into the formula:
[tex]\[ a^2 = 22^2 + 29^2 - 2 \times 22 \times 29 \times \cos(41^\circ) \][/tex]
### Step 3: Calculate each term
1. [tex]\( 22^2 = 484 \)[/tex]
2. [tex]\( 29^2 = 841 \)[/tex]
3. [tex]\( 2 \times 22 \times 29 = 1276 \)[/tex]
4. [tex]\( \cos(41^\circ) \)[/tex] value from trigonometric tables or a calculator.
Now compute the entire expression under the cosine function.
### Step 4: Determine [tex]\( a^2 \)[/tex]
Putting it together:
[tex]\[ a^2 = 484 + 841 - 1276 \times \cos(41^\circ) \][/tex]
[tex]\[ a^2 \approx 484 + 841 - 1276 \times 0.7547 \][/tex]
[tex]\[ a^2 \approx 484 + 841 - 962.25 \][/tex]
[tex]\[ a^2 \approx 361.99 \][/tex]
### Step 5: Compute the value of [tex]\( a \)[/tex]
Taking the square root of both sides:
[tex]\[ a = \sqrt{361.99} \][/tex]
[tex]\[ a \approx 19.03 \][/tex]
### Step 6: Round to the nearest whole number
Finally, we round 19.03 to the nearest whole number:
[tex]\[ n = 19 \][/tex]
So, the value of [tex]\( n \)[/tex] to the nearest whole number is:
19
