Unit A: Geometry

Chapter 3: Trigonometry

Find a Side Length in a Triangle Using the Sine Law

The sine law can be used to find the length of a side if a matching pair and the angle opposite the side are known

Example 2

Use the sine law to find the length of side BC in ΔABC to 1 decimal place.

The matching pair of

∠ C = 119 °

and side c = 7.5 cm and

∠ A = 25 °

are the known values.

Label ΔABC.

The given information in ΔABC is

\begin{array}{rcl} ∠ A & = & 25 ° \\ ∠ C & = & 119 cm \\ c & = & 7.5 cm \end{array}

Recall that

∠ A

is opposite side a and

∠ C

is opposite side c.

To find a side length, the formula is written in the form

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

. When solving, only two of the three fractions in the formula are needed. In solving this triangle for side a,

\frac{b}{\sin B}

is not needed, as neither side b nor

∠ B

is known.

Substitute these values into the sine law to find the length of side a.
Note: side BC is the same side as side a in ΔABC.

\frac{a}{\sin A} = \frac{c}{\sin C} \frac{a}{\sin 25 °} = \frac{7.5}{\sin 119 °}

Multiply both sides by sin 25° to isolate a.

\begin{array}{rcl} \frac{a}{\sin 25 °} \times \sin 25 ° & = & \frac{7.5}{\sin 119 °} \times \sin 25 ° \\ a & = & \frac{7.5}{\sin 119 °} \times \sin 25 ° \\ = & 3.6 cm \end{array}

Note: To calculate

a = \frac{7.5}{\sin 119 °} \times \sin 25 °

on your calculator, do either of the following based on how your calculator works.

Method 1: 7.5

(119)

(25)

Method 2: 7.5

119

The length of side a to 1 decimal place is 3.6 cm.

Example 3

Use the sine law to find the length of side e in ΔDEF to 1 decimal place.

Solution

Label ΔDEF.

In ΔDEF,

∠ D

, which equals 62°, is across from side d, which is unknown.

∠ E

, which equals 75°, is across from side e, which is the unknown side.

∠ F

, which is unknown, is across from side f, which equals 3.9 cm.

When the known values are substituted into the sine law, the following equation results:

\frac{d}{\sin D} = \frac{e}{\sin E} = \frac{f}{\sin F} \frac{d}{\sin 62 °} = \frac{e}{\sin 75 °} = \frac{3.9}{\sin F}

There are no matching pairs. Can the sine law still be used?
To find side e, side d or

∠ F

must be known. In Chapter 1 Lesson 2A, the sum of the angles in a triangle was discussed and can be used to find

∠ F

∠ F

can be calculated using the equation

∠ D + ∠ E + ∠ F = 180 °

\begin{array}{rcl} ∠ D + ∠ E + ∠ F & = & 180 ° \\ 62 ° + 75 ° + ∠ F & = & 180 ° \\ 137 ° + ∠ F & = & 180 ° \\ ∠ F & = & 180 ° - 137 ° \\ = & 43 ° \end{array}

Substitute

∠ F

into the sine law and solve for side e. Since both side f and

∠ F

are known, use this matching pair in the formula.

\begin{array}{rcl} \frac{d}{\sin D} & = & \frac{e}{\sin E} = \frac{f}{\sin F} \\ \frac{d}{\sin 62 °} & = & \frac{e}{\sin 75 °} = \frac{3.9}{\sin 43 °} \\ \frac{e}{\sin 75 °} & = & \frac{3.9}{\sin 43 °} \\ \frac{e}{\sin 75 °} \times \sin 75 ° & = & \frac{3.9}{\sin 43 °} \times \sin 75 ° \\ e & = & \frac{3.9}{\sin 43 °} \times \sin 75 ° \\ = & 5.5 cm \end{array}

Note: When deciding which parts of the sine law to use, ask yourself if you have 3 out of the 4 possible values that need to be substituted in.

\frac{e}{\sin E} = \frac{f}{\sin F}

In this example, we have side f,

∠ F

, and

∠ E

and are only missing side e.

\frac{e}{\sin 75 °} = \frac{3.9}{\sin 43 °}

The length of side e is 5.5 cm.

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