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McGraw Hill Glencoe Algebra 2, 2012

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McGraw Hill Glencoe Algebra 2, 2012 View details

4. Law of Sines

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Exercise 44 Page 820

Think about a right triangle with the measures of two angles and one of the side lengths.

See solution.

Practice makes perfect

Let's imagine that there is a garden at a nursery school in the shape of a right triangle, and we want to cover this garden with grass. We have the following information about the measures of the garden.

Right triangle

To find out how many square meters of grass is needed, we will find the area of this right triangle. However, we are missing the lengths of the legs. With the measures of two angles and the length of the hypotenuse, we can find the missing lengths. To do so we can recall the Law of Sines.

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

Let's first use the second and third ratios to find the value of

c .

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

SubstituteValues

Substitute values

\frac{sin 9 0}{5} = \frac{sin 2 0}{c}

▼

Solve for

c

$LHS \cdot 5 = RHS \cdot 5$

sin 90 = \frac{sin 2 0}{c} \cdot 5

$LHS \cdot c = RHS \cdot c$

sin 90 \cdot c = sin 20 \cdot 5

$LHS / sin 90 = RHS / sin 90$

c = \frac{sin 2 0 \cdot 5}{sin 9 0}

Use a calculator

c = 1.710100 \dots

Round to $1$ decimal place(s)

c \approx 1.7

Then we will find the

m ∠ A

to find the side length opposite to this angle. Let's use the Triangle Angle Sum Theorem.

m ∠ A + 9 0^{\circ} + 2 0^{\circ} = 18 0^{\circ} ⇓ m ∠ A = 7 0^{\circ}

Now, we can find the value of

a

by using the Law of Sines one more time.

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

SubstituteValues

Substitute values

\frac{sin 7 0}{a} = \frac{sin 9 0}{5}

▼

Solve for

a

$LHS \cdot 5 = RHS \cdot 5$

\frac{sin 7 0}{a} \cdot 5 = sin 90

$LHS \cdot a = RHS \cdot a$

sin 70 \cdot 5 = sin 90 \cdot a

$LHS / sin 90 = RHS / sin 90$

\frac{sin 7 0 \cdot 5}{sin 9 0} = a

Rearrange equation

a = \frac{sin 7 0 \cdot 5}{sin 9 0}

Use a calculator

a = 4.698463 \dots

Round to $1$ decimal place(s)

a \approx 4.7

Great! Now we can complete our triangle.

completed triangle

Having the lengths of the legs, we can find the area of this right triangle. Let's do it by using the Area Formula!

Area = \frac{1}{2} \cdot base \cdot height

$base = 4.7$ , $height = 1.7$

Area \approx \frac{1}{2} \cdot (4.7) \cdot (1.7)

Use a calculator

Area \approx 3.995

Round to nearest integer

Area \approx 4

We have found that about

4

square meters grass are needed to cover the garden of the nursery school.

Subchapter links

Check Your Understanding p.818

Practice and Problem Solving p.819-820

H.O.T. Problems p.820

Standardized Test Practice p.821

Spiral Review p.821

Skills Review p.821