Exercise 10 Page 525

For any △ ABC, let the lengths of the sides opposite angles A, B, and C be a, b, and c, respectively.

The Law of Sines relates the sine of each angle to the length of the opposite side. sin A/a=sin B/b=sin C/c Let's use this law to find the values of x and y. We will find them one at a time.

Finding x

Consider the given triangle.

We know that the length of a side is 12 and that the measure of its opposite angle is 38. We also know that the length of the side that is opposite to the angle we want to find is 18. With this information and using the Law of Sines, we can write an equation in terms of x. sin x/18=sin 38/12 Let's solve our equation!

sin x/18=sin 38/12

MultEqn

LHS * 18=RHS* 18

sin x=sin 38/12* 18

To find x we will use the inverse operation of sin, which is sin ^(- 1). sin x=sin 38/12* 18 ⇕ x=sin ^(- 1)(sin 38/12* 18) Finally, we will use a calculator.

x=sin ^(- 1)(sin 38/12* 18)

UseCalc

Use a calculator

x=67.44208077...

RoundDec

Round to 1 decimal place(s)

x≈ 67.4

Finding y

Knowing that x=67.4, we can find the third interior angle using the Triangle Angle Sum Theorem. 180- 38- 67.4= 74.6^(∘) Consider the triangle with the new information.

We know that the length of a side is 12 and that the measure of its opposite angle is 38. We want to find the length of the side that is opposite to the angle whose measure is 74.6. We can use the Law of Sines again! sin 38/12=sin 74.6/y Let's solve the above equation for y using the Cross Product Property.

sin 38/12=sin 74.6/y

CrossMult

Cross multiply

sin 38* y=sin 74.6* 12

DivEqn

.LHS /sin 38.=.RHS /sin 38.

y=sin 74.6* 12/sin 38

UseCalc

Use a calculator

y=18.79140618...

RoundDec

Round to 1 decimal place(s)

y≈ 18.8