Exercise 51 Page 654

Let's begin by drawing △ ABC and labeling the lengths of the sides. We will also color code the opposite angles and sides. It will help us use the Law of Sines and Law of Cosines later.

We will solve △ ABC. This means we will find the values of a, m∠ B, and m∠ C. First, let's find the length of the remaining side and then move on to the measures of the remaining angles.

Finding a

Note that we are given the lengths of two sides of △ ABC and the included angle. Therefore, we can use the Law of Cosines to find a. a^2=b^2+c^2 -2 b c cos A Let's substitute b= 15, c= 24, and m∠ A = 103^(∘) to isolate a.

a^2=b^2+c^2 -2 b c cos A

SubstituteValues

Substitute values

a^2= 15^2+ 24^2 -2 ( 15)( 24) cos 103^(∘)

▼

Solve for a

CalcPow

Calculate power

a^2=225+576-2(15)(24)cos103 ^(∘)

AddTerms

Add terms

a^2=801-2(15)(24)cos103^(∘)

Multiply

Multiply

a^2=801-720cos103^(∘)

SqrtEqn

sqrt(LHS)=sqrt(RHS)

a=sqrt(801-720cos103^(∘))

UseCalc

Use a calculator

a=31.031673 ...

RoundDec

Round to 1 decimal place(s)

a≈ 31.0

The value of a is about 31.0 units. Let's update our diagram.

Finding m ∠ B

Since now we know the lengths of all three sides of △ ABC and the measure of one angle, we can use the Law of Sines. sin A/a =sin B/b Let's substitute m ∠ A = 103^(∘), a ≈ 31.0, and b= 15 to isolate sin B.

sin A/a =sin B/b

SubstituteValues

Substitute values

sin 103^(∘)/31.0 =sin B/15

▼

Solve for sin B

MultEqn

LHS * 15=RHS* 15

15 sin 103^(∘)/31.0 = sin B

RearrangeEqn

Rearrange equation

sin B =15 sin 103^(∘)/31.0

Now we can use the inverse sine ratio to find m ∠ B.

m ∠ B = sin ^(-1) (15 sin 103^(∘)/31.0)

UseCalc

Use a calculator

m ∠ B =28.129719 ...

RoundDec

Round to 1 decimal place(s)

m ∠ B ≈ 28.1^(∘)

Finding m ∠ C

Finally, to find m ∠ C we can use the Triangle Angle Sum Theorem. This tells us that the measures of the angles in a triangle add up to 180^(∘). 103^(∘)+ 28.1^(∘)+ m∠ C = 180^(∘) ⇕ m ∠ C ≈ 48.9^(∘)

Completing the Triangle

With all of the side length and the angle measures, we can complete our diagram.