Exercise 21 Page 675

Let's begin with recalling the Law of Sines. If △ ABC has lengths of a, b, and c and angle measures of A, B, and C, then we can write that the ratios of the sine of an angle to the side opposite to this angle are equal.

In our exercise we are given that Angelina is looking at the Big Dipper through a telescope and from her view, the cup of the constellation forms a triangle. Let's redraw the given diagram.

First we can evaluate the measure of ∠ C. To do this, let's use the Triangle Angle Sum Theorem.

40^(∘)+ 95^(∘)+m∠ C=180^(∘)

AddTerms

Add terms

135^(∘)+m∠ C=180^(∘)

SubEqn

LHS-135^(∘)=RHS-135^(∘)

m∠ C=45^(∘)

The third angle has a measure of 45^(∘). Let's add this information to our diagram.

Since we are asked to evaluate the length of AC, we will create a proportion using the Law of Sines. sin B/AC=sin C/AB Next, we will substitute all values we know using the diagram and solve for AC. To do this, we can use cross multiplication.

sin B/AC=sin C/AB

SubstituteValues

Substitute values

sin 95^(∘)/AC=sin 45^(∘)/2

CrossMult

Cross multiply

2sin 95^(∘)=ACsin 45^(∘)

DivEqn

.LHS /sin 45^(∘).=.RHS /sin 45^(∘).

2sin 95^(∘)/sin 45^(∘)=AC

RearrangeEqn

Rearrange equation

AC=2sin 95^(∘)/sin 45^(∘)

UseCalc

Use a calculator

AC=2.8176...

RoundDec

Round to 1 decimal place(s)

AC≈ 2.8

The length of side AC is approximately 2.8 inches.