Exercise 20 Page 537

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

Let's use this law to find the measure of ∠ K. To do so, we will start by drawing a diagram to illustrate the situation. We will let the measure of ∠ K be x.

We know the length of three sides, 13mi, 10 mi, and 8mi. With this information we want to find one of the interior angles of the triangle. We can use the Law of Cosines to do that.

13^2= 10^2+ 8^2-2( 10)( 8)cos x^(∘)

▼

Solve for cos x^(∘)

CalcPow

Calculate power

169=100+64-2(10)(8)cos x^(∘)

AddTerms

Add terms

169=164-2(10)(8)cos x^(∘)

Multiply

Multiply

169=164-160cos x^(∘)

SubEqn

LHS-164=RHS-164

5=- 160cos x^(∘)

DivEqn

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

5/- 160=cos x^(∘)

UseCalc

Use a calculator

- 0.03125=cos x^(∘)

RearrangeEqn

Rearrange equation

cos x^(∘)=- 0.03125

To find the value of x we will use the inverse operation of cos, which is cos ^(- 1). cos x^(∘)=- 0.03125 ⇕ x=cos ^(- 1)(- 0.03125) Finally, we will use a calculator.

x=cos ^(- 1)(- 0.03125)

UseCalc

Use a calculator

x=91.79078...

RoundDec

Round to 1 decimal place(s)

x≈ 91.8

Therefore, we've found that the measure of ∠ K is approximately 91.8.