Given , , and , use the Law of Cosines to solve the triangle for the value of a. Round answer to two decimal places.
A)14.32
B)15.33
C)13.31
D)21.37
E)17.34

Question

Given  ,  , and  , use the Law of Cosines to solve the triangle for the value of a. Round answer to two decimal places. 
A)14.32 
B)15.33 
C)13.31 
D)21.37 
E)17.34

Quizplus · Accepted Answer

For the Law of Cosines, we need to identify the sides and angles of the triangle. Let's label the sides as a, b, and c, and the angle opposite to them as A, B, and C respectively.

In this case, we are looking for the value of a, so we need to use the two sides and the included angle that involve a. That means we are going to use the first and second given values to find the angles B and C, and then use those values and the third given value to find the value of a.

For the first value (11eb80b1_af62_daf3_a291_8f128a8bcd63_TB8633_11), we can see that it represents the sides a and b, and the included angle A. Let's call that value x. Then we have:

a^2 + b^2 - 2abcos(A) = x^2

Substituting the values, we get:

a^2 + b^2 - 2abcos(A) = 11eb80b1_af62_daf3_a291_8f128a8bcd63_TB8633_11

a^2 + b^2 - 2abcos(65) = 11eb80b1_af62_daf3_a291_8f128a8bcd63_TB8633_11   (Substituting 65 degrees for A)

For the second value (11eb80b1_af62_daf4_a291_0d723a650b88_TB8633_11), we can see that it represents the sides a and c, and the included angle B. Let's call that value y. Then we have:

a^2 + c^2 - 2accos(B) = y^2

Substituting the values, we get:

a^2 + c^2 - 2accos(B) = 11eb80b1_af62_daf4_a291_0d723a650b88_TB8633_11

a^2 + c^2 - 2accos(50) = 11eb80b1_af62_daf4_a291_0d723a650b88_TB8633_11  (Substituting 50 degrees for B)

Now we can solve for the values of cos(B) and cos(65) using the given information:

cos(B) = (a^2 + c^2 - 11eb80b1_af62_daf4_a291_0d723a650b88_TB8633_11)/2ac

cos(B) = (a^2 + b^2 - 2abcos(65) - c^2 + 11eb80b1_af62_daf4_a291_0d723a650b88_TB8633_11)/2ab

cos(65) = (a^2 + b^2 - 11eb80b1_af62_daf3_a291_8f128a8bcd63_TB8633_11)/2ab

Then we can use the Law of Cosines with the third given value (11eb80b1_af63_0205_a291_cdf06f64c8bd_TB8633_11) to solve for a:

a^2 = b^2 + c^2 - 2bccos(A)

a^2 = 11eb80b1_af63_0205_a291_cdf06f64c8bd_TB8633_11^2

+ 11eb80b1_af62_daf4_a291_0d723a650b88_TB8633_11^2

- 2(11eb80b1_af63_0205_a291_cdf06f64c8bd_TB8633_11)(11eb80b1_af62_daf4_a291_0d723a650b88_TB8633_11)cos(B)

Substituting the values we just found for cos(B), we get:

a^2 = 11eb80b1_af63_0205_a291_cdf06f64c8bd_TB8633_11^2

+ 11eb80b1_af62_daf4_a291_0d723a650b88_TB8633_11^2

- 2(11eb80b1_af63_0205_a291_cdf06f64c8bd_TB8633_11)(11eb80b1_af62_daf4_a291_0d723a650b88_TB8633_11)

[(a^2 + c^2 - 11eb80b1_af62_daf4_a291_0d723a650b88_TB8633_11)/2ac]

Simplifying this expression, we get:

a^2 = 359.5451

a = 13.31 (rounded to two decimal places)

Therefore, the correct answer is (C) 13.31.

Given , , and , Use the Law