SOLUTION: Find the measure of each angle in the triangle shown in the figure below. Round to the nearest tenth. https://www.webassign.net/waplots/9/d/55e00c7827afd9b2f4b22b74cb8afb.gif

Algebra ->  Trigonometry-basics -> SOLUTION: Find the measure of each angle in the triangle shown in the figure below. Round to the nearest tenth. https://www.webassign.net/waplots/9/d/55e00c7827afd9b2f4b22b74cb8afb.gif      Log On


   

Question 1204600: Find the measure of each angle in the triangle shown in the figure below. Round to the nearest tenth.
https://www.webassign.net/waplots/9/d/55e00c7827afd9b2f4b22b74cb8afb.gif
Found 2 solutions by MathLover1, math_tutor2020:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Side+a+=+7
Side b+=+12
Side c+=+10

A+=+cos%5E-1%28%28b%5E2+%2B+c%5E2+-+a%5E2%29%2F2bc%29
A+=+cos%5E-1%28%2812%5E2+%2B+10%5E2+-+7%5E2%29%2F%282%2A12%2A10%29%29
A+=+35.67°


B+=+cos%5E-1%28%28a%5E2+%2B+c%5E2+-+b%5E2%29%2F2ac%29
B+=+cos%5E-1%28%287%5E2+%2B+10%5E2+-+12%5E2%29%2F%282%2A7%2A10%29%29
B+=+87.95°

C+=+cos%5E-1%28%28a%5E2+%2B+b%5E2+-+c%5E2%29%2F2ab%29
C+=+cos%5E-1%28%287%5E2+%2B+12%5E2+-+10%5E2%29%2F%282%2A7%2A12%29%29
C+=+56.4°

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!


Given Sides
a = 7
b = 12
c = 10
We have the SSS case since we know all three sides of the triangle.

Whenever we have the SAS case or SSS case, we can use the Law of Cosines to solve the triangle.
The phrasing "solve the triangle" specifically refers to the idea of finding all three sides and the measure of all three angles.

Let's use the law of cosines to find angle A.
a^2 = b^2 + c^2 - 2*b*c*cos(A)
7^2 = 12^2 + 10^2 - 2*12*10*cos(A)
49 = 144 + 100 - 240*cos(A)
49 - 144 - 100 = -240*cos(A)
-195 = -240*cos(A)
cos(A) = -195/(-240)
cos(A) = 0.8125 exactly
A = arccos(0.8125)
A = 35.6591 approximately
A = 35.7 degrees approximately

Similar steps will be followed to find angles B and C.
Here are the equations to solve using the Law of Cosines
b^2 = a^2+c^2 - 2*a*c*cos(B)
and
c^2 = a^2+b^2 - 2*a*b*cos(C)

I'll skip steps since they are similar to what is shown above.
Here are the results you should get
B = 88 degrees approximately
C = 56.4 degrees approximately

GeoGebra is a useful tool to verify the answers are correct.


Unfortunately A+B+C = 35.7+88+56.4 = 180.1 which shows we have a bit of rounding error.

The slightly more accurate angle values are
A = 35.6591
B = 87.9533
C = 56.3876
and,
A+B+C = 35.6591+87.9533+56.3876 = 180