In triangle LMN, angle LNM = 114 degrees, LM = 123 mm and MN = 88mm. Calculate angle LMN, to the nearest degree.
use law of sines to find NLM
180º - LNM - NLM = LMN
To calculate angle LMN, we can use the Law of Cosines. This law relates the lengths of the sides of a triangle to the cosine of one of its angles. The formula is as follows:
c^2 = a^2 + b^2 - 2ab * cos(C)
Where:
- c is the side opposite angle C
- a and b are the lengths of the other two sides
- C is the angle opposite side c
In this case, we want to find angle LMN (which is angle C), given the lengths of the sides LM and MN.
Let's plug in the given values:
LM = 123 mm
MN = 88 mm
We need to find the length of side LN (opposite to angle LMN).
Using the Law of Cosines:
LN^2 = LM^2 + MN^2 - 2 * LM * MN * cos(LMN)
Substituting the given values:
LN^2 = 123^2 + 88^2 - 2 * 123 * 88 * cos(114)
Now, we will solve this equation to find the length of side LN.
LN^2 = 15129 + 7744 - 21624 * cos(114)
LN^2 = 22873 - 21624 * cos(114)
Using the square root, we find:
LN = sqrt(22873 - 21624 * cos(114))
Now that we have determined the length of side LN, we can use the Law of Sines to find angle LMN:
sin(LMN) = MN / LN
Substituting the given values:
sin(LMN) = 88 / sqrt(22873 - 21624 * cos(114))
Finally, calculate angle LMN by taking the inverse sine (or arcsin) of both sides:
LMN = arcsin(88 / sqrt(22873 - 21624 * cos(114)))
Calculating this expression will give us the value of angle LMN, to the nearest degree.