This is the question:
In triangle LMN, angle LMN = 42 degrees, LN = 32mm, and LM = 42 mm. Calculate all possible measures of angle MLN to the nearest degree. Use the sine ratio and show your work.
just crank it out. We know that
sin42°/32 = sinN/42
sinN = 0.878
N = 61.43° or 118.57°
That means that angle L is 76.57° or 19.43°
Now use those to find MN
To calculate all possible measures of angle MLN using the sine ratio, we need to use the formula:
sin(Angle) = opposite/hypotenuse.
In triangle LMN, LN is the hypotenuse and LM is the side opposite to angle MLM.
Given that LN = 32mm and LM = 42mm, we can calculate the sine of angle MLM as follows:
sin(MLM) = LM/LN = 42/32
Now, we can find the actual measure of angle MLM by using the inverse sine function:
MLM = sin^(-1)(42/32)
Using a calculator, we find MLM ≈ 54.41 degrees.
Since we're looking for all possible measures of angle MLN, we have two scenarios:
1. When angle MLN is acute (less than 90 degrees): In this case, angle MLN must be larger than angle MLM. We can find the acute angle by subtracting MLM from 180 degrees:
Acute angle MLN = 180 - MLM
Acute angle MLN ≈ 180 - 54.41 ≈ 125.59 degrees.
2. When angle MLN is obtuse (greater than 90 degrees): In this case, angle MLN must be larger than 90 degrees but smaller than 180 degrees. Therefore, there is only one obtuse angle possible:
Obtuse angle MLN = 180 - acute angle MLN
Obtuse angle MLN ≈ 180 - 125.59 ≈ 54.41 degrees.
Therefore, the possible measures of angle MLN to the nearest degree are approximately 54 degrees (acute) and 125 degrees (obtuse).
To find the measure of angle MLN in triangle LMN using the sine ratio, we can use the following formula:
sin(angle MLN) = opposite/hypotenuse
Given that LN is the hypotenuse and LM is the side opposite to angle MLN, we can substitute the values:
sin(angle MLN) = LM/LN
sin(angle MLN) = 42/32
Now, let's calculate the value of sin(angle MLN):
sin(angle MLN) ≈ 0.763
To find the measure of angle MLN, we need to take the inverse sine (also known as arcsin) of the value obtained:
angle MLN ≈ arcsin(0.763)
Using a calculator, we find:
angle MLN ≈ 49.3 degrees
Therefore, the possible measure of angle MLN to the nearest degree is approximately 49 degrees.