The law of sines is a useful equation that relates the sides of a triangle to the sine of the corresponding angles and can be used to solve for missing values in an oblique triangle, or triangle that does not include a right angle.

\displaystyle \dfrac{\sin{A}}{a}=\dfrac{\sin{B}}{b}=\dfrac{\sin{C}}{c}

\bm{A,\,B,\,C}

the measures of the angles in the triangle

\bm{a,\,b,\,c}

the sides in the triangle

Triangle A B C. Side B C, the side opposite angle A, has a length of lowercase a. Side A C, the side opposite angle B, has a length of lowercase b. Side A B, the side opposite angle C, has a length of lowercase c. Arrows from angle A to lowercase a, from angle B to lowercase B, and from angle C to lowercase C, are drawn inside the triangle.

In order to apply the law of sines, we must be given an angle and its opposite side plus one additional side or angle. We will only use two proportions at a time to solve for missing values.

When solving a triangle given two angles and a side, we are guaranteed one unique solution.

However, when using the law of sines to solve a triangle given two sides and the non-included angle we may not always have a valid solution.

Worked examples

Example 1

Consider the triangle shown in the figure:

Triangle A B C. Angle B measures 85 degrees and angle C measures 43.2 degrees. Side A C, the side opposite angle B, has a length of 7.8. Side B C, the side opposite angle A, has a length of x.

Write the proportions that relate the sides and angles of the triangle using the law of sines.

Solution

\frac{\sin{A}}{x}=\frac{\sin{85\degree}}{7.8}=\frac{\sin{43.2\degree}}{AB}

Reflection

There is no variable label on side \overline{AB} but we can always refer to the measure using the two endpoints.

Solve for x.

Approach

In order to solve for a missing side using the law of sines you need an angle and its opposite side, plus the oppposite angle. Since we know both m\angle{B} and the length of side \overline{AC} we can solve for the value of x.

It may appear at first glance that we don't have enough information because we don't know m\angle{A}, but by applying the triangle sum theorem we can solve for m\angle{A} in order to find x.

Solution

Using the triangle sum theorem: m\angle{A}=180\degree-(85\degree+43.2\degree)=51.8\degree

\displaystyle \frac{a}{\sin{A}}	\displaystyle =	\displaystyle \frac{b}{\sin{B}}	Law of sines
\displaystyle \frac{x}{\sin{51.8\degree}}	\displaystyle =	\displaystyle \frac{7.8}{\sin{85\degree}}	Substitute known values
\displaystyle x	\displaystyle =	\displaystyle \sin{51.8\degree}\left(\frac{7.8}{\sin{85\degree}}\right)	Multiply both sides by \sin{51.8\degree}
\displaystyle x	\displaystyle =	\displaystyle 6.15	Evaluate on a calculator

Reflection

The law of sines can be reciprocated to solve for any missing value. You can see in this example instead of \dfrac{\sin{A}}{a}=\dfrac{\sin{B}}{b} we used \dfrac{a}{\sin{A}}=\dfrac{b}{\sin{B}}.

Example 2

Solve for all of the missing sides and angles in a triangle where a=7.1 \, \text{ft}, c=5.3 \, \text{ft}, and m\angle A=61\degree.

Approach

Draw and label a diagram of the triangle; it does not need to be drawn to scale.

From this diagram we can see that we were given two sides and their non-included angle.

We are given a pair of opposite sides and angles with side a and angle A in addition to side c. We will set up the equation: \frac{\sin{A}}{a}=\frac{\sin{C}}{c} to find m\angle{C} to start.

Solution

\displaystyle \frac{\sin{A}}{a}	\displaystyle =	\displaystyle \frac{\sin{C}}{c}	Law of sines
\displaystyle \frac{\sin{61 \degree}}{7.1}	\displaystyle =	\displaystyle \frac{\sin{C}}{5.3}	Substitute given values
\displaystyle 5.3\left(\frac{\sin{61 \degree}}{7.1}\right)	\displaystyle =	\displaystyle \sin{C}	Multiplication property of equality
\displaystyle \sin^{-1}\left(5.3\left(\frac{\sin{61 \degree}}{7.1}\right)\right)	\displaystyle =	\displaystyle C	Apply the inverse sine to both sides
\displaystyle 40.76\degree	\displaystyle =	\displaystyle C	Evaluate

Next, we can find m \angle B using the triangle angle sum. m\angle B=180\degree - 61 \degree - 40.76 \degree = 78.24 \degree

Now that we have the measure of angle B, we can find the side b using the law of sines again.

\displaystyle \frac{a}{\sin{A}}	\displaystyle =	\displaystyle \frac{b}{\sin{B}}	Law of sines
\displaystyle \frac{7.1}{\sin{61\degree}}	\displaystyle =	\displaystyle \frac{b}{\sin{78.24\degree}}	Substitute given values
\displaystyle \sin{78.24\degree}\left(\frac{7.1}{\sin{61\degree}}\right)	\displaystyle =	\displaystyle b	Multiplication property of equality
\displaystyle 7.95	\displaystyle =	\displaystyle b	Evaluate

Now we know the measure of all of the sides and angles of this triangle. We can do a sketch to see if the side measurements are reasonable.

Reflection

Notice that the smallest angle is opposite the shortest side and that the largest angle is opposite the longest side. This will always be the case.

Outcomes

M3.N.Q.A.1.A

Choose and interpret the scale and the origin in graphs and data displays.

M3.G.SRT.C.5

Solve triangles.*

M3.G.SRT.C.5.C

Use the Law of Sines and Law of Cosines to solve non-right triangles in a real-world context.

M3.MP1

Make sense of problems and persevere in solving them.

M3.MP2

Reason abstractly and quantitatively.

M3.MP3

Construct viable arguments and critique the reasoning of others.

M3.MP4

Model with mathematics.

M3.MP5

Use appropriate tools strategically.

M3.MP6

Attend to precision.

M3.MP7

Look for and make use of structure.

M3.MP8

Look for and express regularity in repeated reasoning.

4.07 Law of sines

Concept summary

Worked examples

Example 1

Solution

Reflection

Approach

Solution

Reflection

Example 2

Approach

Solution

Reflection

Outcomes

M3.N.Q.A.1.A

M3.G.SRT.C.5

M3.G.SRT.C.5.C

M3.MP1

M3.MP2

M3.MP3

M3.MP4

M3.MP5

M3.MP6

M3.MP7

M3.MP8

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