use the law of cosines to solve the triangle a=55, c=72, b=25
To solve the triangle using the law of cosines, we need to find the measure of angle A and side b.
The law of cosines states:
c^2 = a^2 + b^2 - 2ab * cos(C)
where c is the given side length opposite angle C, and a and b are the other two sides.
Given:
a = 55
c = 72
b = 25
Let's solve for angle C first.
c^2 = a^2 + b^2 - 2ab * cos(C)
Substituting the given values:
72^2 = 55^2 + 25^2 - 2 * 55 * 25 * cos(C)
5184 = 3025 + 625 - 2750 * cos(C)
Now, simplify the equation:
5184 = 3650 - 2750 * cos(C)
Move all terms to one side to form a quadratic equation:
0 = -2750 * cos(C) - 1834
Let's solve for cos(C) by dividing both sides of the equation by -2750:
cos(C) = 1834 / 2750
Calculate the value of cos(C):
cos(C) ≈ 0.666909
Now, we need to find the measure of angle C. To do this, we can use the inverse cosine function (cos⁻¹) or arccos on a calculator.
C ≈ cos⁻¹(0.666909)
C ≈ 47.7 degrees
Now that we have the measure of angle C, we can use the law of cosines again to find the length of side b.
b^2 = a^2 + c^2 - 2ac * cos(B)
Substituting the values we know:
b^2 = 55^2 + 72^2 - 2 * 55 * 72 * cos(B)
Simplify the equation:
b^2 = 3025 + 5184 - 7920 * cos(B)
Combine like terms:
b^2 = 8209 - 7920 * cos(B)
Now, substitute the value of cos(C):
b^2 = 8209 - 7920 * 0.666909
Calculate the value of b^2:
b^2 ≈ 8209 - 5280.7392
b^2 ≈ 2928.2608
Finally, we can find the length of side b by taking the square root of b^2:
b ≈ √2928.2608
b ≈ 54.07
Therefore, the length of side b is approximately 54.07.
cosB = (a^2 + c^2 - b^2) / 2ac,
cosB = ((55)^2 + (72)^2 - (25)^2/7920 = 7584 / 7920 = 0.9576.
B = 16.7 deg.