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McGraw Hill Glencoe Algebra 2, 2012 View details

4. Law of Sines

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Exercise 40 Page 820

Practice makes perfect

a We are given a diagram showing the Bermuda Triangle. We want to find the distance between Miami and Bermuda. Consider the diagram. We will call the missing distance

c .

We can find the distance using the Law of Sines. However, to do so we first need to know the measure of the angle formed at San Juan. We will start by finding this angle measure.

Finding the Measure of Angle Formed at San Juan

Consider our diagram.

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

Let's substitute

a = 965,

m ∠ A = 53,

and

b = 1038 .

We will call the angle formed at Bermuda

x .

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

SubstituteValues

Substitute values

\frac{sin 5 3}{9 6 5} = \frac{sin x}{1 0 3 8}

▼

Solve for

sin x

MultEqn

$LHS \cdot 1038 = RHS \cdot 1038$

\frac{1 0 3 8 sin 5 3}{9 6 5} = sin x

RearrangeEqn

Rearrange equation

sin x = \frac{1 0 3 8 sin 5 3}{9 6 5}

Now we can use the inverse sine ratio to find

x .

x = sin^{- 1} \frac{1 0 3 8 sin 5 3}{9 6 5}

UseCalc

Use a calculator

x \approx 59.210131 \dots

RoundInt

Round to nearest integer

x \approx 59

Knowing the measure of the angle formed at Bermuda, we can use the Triangle Angle Sum Theorem to find the measure of the angle formed at San Juan. It tells us that the sum of the angle measures in a triangle is

18 0^{\circ} .

Let's call the missing angle

y .

y + 5 3^{\circ} + 5 9^{\circ} = 18 0^{\circ} \Rightarrow y \approx 6 9^{\circ}

Finding the Distance

Let's add the obtained angle measures to our diagram.

Since we know the measure of two angles and the length of the side opposite to one of them, we can use the Law of Sines to find the desired value.

\frac{sin B}{b} = \frac{sin C}{c}

We can substitute

m ∠ B = 59,

m ∠ C = 69,

and

b = 1038

and solve for the desired value.

\frac{sin B}{b} = \frac{sin C}{c}

SubstituteValues

Substitute values

\frac{sin 5 9}{1 0 3 8} = \frac{sin 6 9}{c}

▼

Solve for

c

MultEqn

$LHS \cdot c = RHS \cdot c$

\frac{sin 5 9}{1 0 3 8} \cdot c = sin 69

MultEqn

$LHS \cdot 1038 = RHS \cdot 1038$

sin 59 \cdot c = 1038 sin 69

DivEqn

$LHS / sin 59 = RHS / sin 59$

c = \frac{1 0 3 8 sin 6 9}{sin 5 9}

UseCalc

Use a calculator

c = 1130.533657 \dots

RoundInt

Round to nearest integer

c \approx 1131

The distance between Miami and Bermuda is approximately

1131

miles.

b We now want to find the approximate area of the Bermuda Triangle. Let's consider our diagram once again and add all the obtained information.

To calculate its area of our triangle, we can use the formula for the area of a triangle using sine.

Area Area Area = \frac{1}{2} b c sin A = \frac{1}{2} a c sin B = \frac{1}{2} a b sin C

Since we know all of the side lengths and all of the angle measures in our triangle, we can use whichever version of the formula we want. For simplicity, let's choose the first one. We will substitute