For any $△ A B C,$ let the lengths of the sides opposite angles $A,$ $B,$ and $C$ be $a,$ $b,$ and $c,$ respectively.

The Law of Sines relates the sine of each angle to the length of the opposite side.

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

Let's start by finding

m ∠ N .

Then, we will use the above law to find the values of

p

and

q .

We will find them one at a time.

Consider the triangle that satisfies the given conditions.

From the Triangle Angle Sum Theorem we know that the sum of the angles in a triangle is equal to

18 0^{\circ} .

With this information we can find

N .

m ∠ N + 5 7^{\circ} + 10 9^{\circ} = 18 0^{\circ} ⇕ m ∠ N = 1 4^{\circ}

Let's mark the measure of $∠ N$ on the graph.

We know that the length of a side is

22

and that the measure of its opposite angle is

1 4^{\circ} .

We also know that the measure of the angle that is opposite to the side we want to find is

10 9^{\circ} .

With this information and using the Law of Sines, we can write an equation in terms of

p .

\frac{sin 1 4 ^{\circ}}{2 2} = \frac{sin 1 0 9 ^{\circ}}{p}

Let's solve the above equation for

p

\frac{sin 1 4 ^{\circ}}{2 2} = \frac{sin 1 0 9 ^{\circ}}{p}

Cross multiply

sin 1 4^{\circ} \cdot p = sin 10 9^{\circ} \cdot 22

$LHS / sin 1 4^{\circ} = RHS / sin 1 4^{\circ}$

p = \frac{sin 1 0 9 ^{\circ} \cdot 2 2}{sin 1 4 ^{\circ}}

Use a calculator

p = 85.983985 \dots

Round to $1$ decimal place(s)

p \approx 86.0

Consider the triangle with the new information.

We know that the length of a side is

22

and that the measure of its opposite angle is

1 4^{\circ} .

We want to find the length of the side that is opposite to the angle whose measure is

5 7^{\circ} .

We can use the Law of Sines again!

\frac{sin 2 2 ^{\circ}}{1 4} = \frac{sin 5 7 ^{\circ}}{q}

Let's solve the above equation for

q

using the Cross Product Property.

\frac{sin 2 2 ^{\circ}}{1 4} = \frac{sin 5 7 ^{\circ}}{q}

Cross multiply

sin 2 2^{\circ} \cdot q = sin 5 7^{\circ} \cdot 14

$LHS / sin 2 2^{\circ} = RHS / sin 2 2^{\circ}$

q = \frac{sin 5 7 ^{\circ} \cdot 1 4}{sin 2 2 ^{\circ}}

Use a calculator

q = 31.343249 \dots

Round to $1$ decimal place(s)

q \approx 31.3

Exercise 26 Page 819

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