Question

Solve the triangle. (Standard notation for triangle ABC

is used throughout. Round your answer to two decimal places.)
b = 18.8, c = 26.5, B = 27.9°
larger valueA A=
C=
a=
smaller value for A
A=
C=

Answer 1

Explanation:

Given:

b = 18.8

c = 26.5

B = 27.9°

To find angle A, we can use the Law of Sines:

sin(A)/a = sin(B)/b

We know B and b, so we can substitute the values:

sin(A)/a = sin(27.9°)/18.8

Now, we can solve for sin(A):

sin(A) = (sin(27.9°)/18.8) * a

To find the value of a, we can use the Law of Cosines:

a^2 = b^2 + c^2 - 2bc*cos(B)

Substituting the given values:

a^2 = 18.8^2 + 26.5^2 - 2 * 18.8 * 26.5 * cos(27.9°)

Now, we can solve for a:

a = sqrt(18.8^2 + 26.5^2 - 2 * 18.8 * 26.5 * cos(27.9°))

Using the Law of Sines again, we can find angle C:

sin(C)/c = sin(B)/b

Substituting the known values:

sin(C)/26.5 = sin(27.9°)/18.8

Now, we can solve for sin(C):

sin(C) = (sin(27.9°)/18.8) * 26.5

Finally, we can solve for angle C:

C = arcsin((sin(27.9°)/18.8) * 26.5)

To find the smaller value for angle A, we can subtract angle B and angle C from 180°:

A = 180° - B - C

Now, we can calculate the values:

A ≈ 180° - 27.9° - arcsin((sin(27.9°)/18.8) * 26.5)

C ≈ arcsin((sin(27.9°)/18.8) * 26.5)

a ≈ sqrt(18.8^2 + 26.5^2 - 2 * 18.8 * 26.5 * cos(27.9°))

Please note that the final numerical calculation is required to provide the exact values for A, C, and a.