Exercise 37 Page 820

We are given a diagram showing the locations of Hilo, Kailua, and Captain Cook in Hawaii. We want to find the angle measure formed at Hilo.

Since we do not know the length of the side opposite to the angle formed at Hilo, we cannot yet find its measure. However, notice that we can use the Law of Sines to find the measure of angle formed at Kailua. Then we will use the Triangle Angle Sum Theorem to find the desired angle measure.

Let's substitute

a = 57,

m ∠ A = 80,

and

b = 55 .

We will call the angle at Kailua

x .

\frac{sin A}{a} = \frac{sin B}{b}

SubstituteValues

Substitute values

\frac{sin 8 0}{5 7} = \frac{sin x}{5 5}

▼

Solve for

sin x

MultEqn

$LHS \cdot 55 = RHS \cdot 55$

\frac{5 5 sin 8 0}{5 7} = sin x

RearrangeEqn

Rearrange equation

sin x = \frac{5 5 sin 8 0}{5 7}

Now we can use the inverse sine ratio to find

x .

x = sin^{- 1} \frac{5 5 sin 8 0}{5 7}

UseCalc

Use a calculator

x \approx 71.851626 \dots

RoundInt

Round to nearest integer

x \approx 72

Knowing the measure of the angle formed at Kailua, we can use the Triangle Angle Sum Theorem to find the measure of the angle formed at Hilo. It tells us that the sum of the angle measures in a traingle is

18 0^{\circ} .

Let's call the missing angle

y .

y + 8 0^{\circ} + 7 2^{\circ} = 18 0^{\circ} \Leftrightarrow y \approx 2 8^{\circ}