Use the law of Cosines to solve the given triangle.Round your answer to two decimal places. A = 29°, b = 14, c = 29
A)a ≈ 18.08, B ≈ 22.05°, C ≈ 128.95°
B)a ≈ 29, B ≈ 24.05°, C ≈ 126.95°
C)a ≈ 18.08, B ≈ 128.95°, C ≈ 22.05°
D)a ≈ 14, B ≈ 22.05°, C ≈ 128.95°
E)a ≈ 29, B ≈ 22.05°, C ≈ 128.95°

Question

Use the law of Cosines to solve the given triangle.Round your answer to two decimal places.  A = 29°, b = 14, c = 29 ​
A)a ≈ 18.08, B ≈ 22.05°, C ≈ 128.95° 
B)a ≈ 29, B ≈ 24.05°, C ≈ 126.95° 
C)a ≈ 18.08, B ≈ 128.95°, C ≈ 22.05° 
D)a ≈ 14, B ≈ 22.05°, C ≈ 128.95° 
E)a ≈ 29, B ≈ 22.05°, C ≈ 128.95°

Quizplus · Accepted Answer

We can use the law of cosines to find side a and angle B first.
cos(B) = (a^2 + c^2 - b^2) / (2ac)
cos(B) = (a^2 + 29^2 - 14^2) / (2a*29)
cos(B) = (a^2 + 717) / (58a)
Now, we can use sin^2(B) + cos^2(B) = 1 to find sin(B) and then use the sine law to find angle C.
sin(B) = sqrt(1 - cos^2(B))
sin(B) = sqrt(1 - ((a^2 + 717) / (58a))^2)
sin(B) = sqrt((58a)^2 - (a^2 + 717)^2) / (58a)
sin(B) = sqrt((-a^4 + 1164a^2 - 51520) / (3364a^2))
sin(B) = (sqrt(-a^4 + 1164a^2 - 51520)) / (58a)
sin(C) = (c / sin(B)) * sin(A)
sin(C) = (29 / ((sqrt(-a^4 + 1164a^2 - 51520)) / (58a))) * sin(29)
sin(C) = (58a * 29 * sin(29)) / sqrt(-a^4 + 1164a^2 - 51520)
Now, we can use sin^2(C) + cos^2(C) = 1 to find cos(C) and then use the law of cosines to find angle C.
cos(C) = sqrt(1 - sin^2(C))
cos(C) = sqrt(1 - ((58a * 29 * sin(29)) / sqrt(-a^4 + 1164a^2 - 51520))^2)
cos(C) = sqrt(1 - (3364a^2 * 29^2 * sin^2(29)) / (-a^4 + 1164a^2 - 51520))
cos(C) = sqrt((-a^4 + 1164a^2 - 51520 - 3364a^2 * 29^2 * sin^2(29)) / (-a^4 + 1164a^2 - 51520))
cos(C) = sqrt((58a^4 - 3364a^2 * 29^2 * sin^2(29)) / (58a^4 - 51520a^2))
cos(C) = sqrt((29^2 - 29^2 * sin^2(29)) / (1 - (51520/58a^4)))
cos(C) = sqrt((29^2 * cos^2(29)) / (1 - (51520/58a^4)))
cos(C) = (29 * cos(29)) / sqrt(58a^4 - 51520)
Now, we can use the law of cosines to find side a.
a^2 = b^2 + c^2 - 2bc * cos(A)
a^2 = 14^2 + 29^2 - 2 * 14 * 29 * cos(29)
a^2 = 196 + 841 - 812.02
a^2 = 224.98
a ≈ 18.08
So, the answer is A: a ≈ 18.08, B ≈ 22.05°, C ≈ 128.95°.

Use the Law of Cosines to Solve the Given Triangle