Question 312421: in a triangle, the sides measure 3,5, and 7 what is the measure, in degrees of the largest angle?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! Let the triangle be ABC with side a opposite angle A, side b opposite angle B, side c opposite angle C.
Let:
side a = 3
side b = 5
side c = 7
Since you know all 3 sides, then you can use the law of cosines to find one of the angles.
Use the formula:
c^2 = a^2 + b^2 - 2*a*b*cos(C)
This becomes:
7^2 = 3^2 + 5^2 - 2*3*5*cos(C)
Simplify to get:
49 = 34 - 30*cos(C)
Subtract 34 from both sides of this equation to get:
15 = -30*cos(C)
Solve for cos(C) to get:
cos(C) = -.5
Solve for C to get:
C = 120 degrees.
You can use the law of cosines again to get another of the angles or you can use the law of sines.
If you use the law of cosines, then you would solve for:
b^2 = a^2 + c^2 - 2*a*c*cos(B)
That would get you:
25 = 9 + 49 - 42*cos(B) which would get you:
-42*cos(B) = -33 which would get you:
cos(B) = .785714286 which would get you:
B = 38.2132107
If you used the law of sines to find angle B, you would use the formula:
b/sin(B) = c/sin(C) which would become:
5/sin(B) = 7/sin(120) which would become:
5/sin(B) = 7/.866025404 which would become:
5/sin(B) = 8.082903769
Multiply both sides by sin(B) and divide both sides by 8.082903769 to get:
sin(B) = 5/8.082903769 which would become:
sin(B) = .618589574 which would become:
B = 38.2132107 degrees.
Since we got the second angle using the law of sines and using the law of cosines, then we're probably on the right track.
Since the sum of the angles of a triangle is equal to 180 degrees, this makes angle A equal to 21.7867893 degrees.
The three angles of your triangle are:
A = 21.7867893 degrees
B = 38.2132107 degrees
C = 120 degrees
The largest angle is angle C.
You can find the third angle in 3 ways.
1. Sum of angles of a triangle = 180 degrees.
2. Law of Sines
3. Law of Cosines
If you did it correctly, all 3 ways will get you the same answer.
Using the law of sines, the ratio for all sides divided by the sin of their angles has to be the same.
Take any one of the sides divided by the sine of their angle to get the ratio.
Using c/sin(c), we get 7/sin(120) which becomes 8.082903769
Since side / sine of the angle must equal the ratio, then sine of the angle must equal side / ratio.
3/8.082903769 gets you .3471153744 which get you an angle of 21.789...
5/8.082903769 gets you .618589574 which gets you an angle of 38.21...
All 3 angles follow the law of sines.
The third angle can also be found using the law of cosines by using the formula:
a^2 = b^2 + c^2 - 2*b*c*cos(A)
This becomes:
3^2 = 5^2 + 7^2 - 2*5*7*cos(A) which simplifies to:
9 = 25 + 49 - 70*cos(A) which simplifies further to:
-70*cos(A) = -65 which simplifies further to:
cos(A) = -65/-70 = .928571429 which gets you an angle of 21.7867893
All 3 ways get you the same angles as they should.
It appears that the largest angle is opposite the largest side, as it should in a triangle.
Solving for all 3 angles confirms that.
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