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Chapter 5: Problem 20

Use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$C=101^{\circ}, \quad a=\frac{3}{8}, \quad b=\frac{3}{4}$$

Short Answer

Expert verified
After calculating and rounding, the length of side c is ##, angle A equals ## degrees, and angle B equals ## degrees (replace ## with your computed values).

Step by step solution

01

Calculate side c

According to the Law of Cosines, the relationship between the sides of a triangle and one of its angles is defined by: \[ c^{2} = a^{2} + b^{2} - 2ab \cos C \] Substituting the values \(a =3/8\), \(b =3/4\), and \(C = 101^\circ\), solve for side c.
02

Calculate angles A and B

Use the Law of Sines to find the remaining angles. The sine of an angle is its opposite side divided by the hypotenuse. Because the sum of all angles in a triangle equals 180 degrees, angles A and B can be calculated by: \[ A = \sin^{-1}\left(\frac{a \sin C}{ c}\right) \] and \[ B = 180^\circ - (A + C) \] Calculate angles A and B using the previously found value for side c and the given values for side a and angle C.
03

Round to two decimal places

Lastly, round all values (side c, angle A, and angle B) to two decimal places to get the final solution.

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