Exercise 9 Page 539

For any △ ABC, the Law of Cosines relates the cosine of each angle to the side lengths of the triangle.

To find the missing angle measure, we will start by drawing a diagram to illustrate the situation.

We know that the lengths of XY, YZ, and ZX are 12, 10, and 8, respectively. With this information and using the Law of Cosines, we can write an equation to find cos Z.

12^2= 10^2+ 8^2-2( 10)( 8)cos Z

▼

Solve for cos Z

CalcPow

Calculate power

144=100+64-2(10)(8)cos Z

AddTerms

Add terms

144=164-2(10)(8)cos Z

Multiply

Multiply

144=164-160cos Z

SubEqn

LHS-164=RHS-164

- 20 = - 160cos Z

DivEqn

.LHS /(- 160).=.RHS /(- 160).

20/160=cos Z

CalcQuot

Calculate quotient

0.125=cos Z

To find m∠ Z, we will use the inverse operation of cos, which is cos ^(- 1). 0.125=cos Z ⇔ cos ^(- 1)0.125=Z Finally, we will use a calculator.

cos ^(- 1)0.0125=Z

UseCalc

Use a calculator

82.819215... ^(∘) =Z

RoundDec

Round to 1 decimal place(s)

82.8^(∘)≈ Z

RearrangeEqn

Rearrange equation

Z≈ 82.8^(∘)

We found that Z≈ 82.8, which corresponds to answer D.