Law of sines

Here is everything you need to know about the Law of Sines. You will learn what it is and how to use it to find the missing sides and angles of a triangle.

Students first learn how to work with the law of sines in geometry and expand that knowledge as they progress through algebra 2 and precalculus.

What is the law of sines?

The Law of Sines (or the sine rule) is a proportional relationship between the size of an angle in a triangle and its opposite side.

The Law of Sines is used to find the missing sides and missing angles of a triangle.

Recall that SOHCAHTOA is used to find missing sides and angles in right triangles (right-angled triangles).

The Law of Sines can be used with any oblique triangle and also with right triangles.

The proportional relationship is:

\cfrac{\sin(A)}{a}=\cfrac{\sin(B)}{b}=\cfrac{\sin(C)}{c}

Notice that angle A (vertex A ) is across from side a, angle B (vertex B ) is across side b, and angle C (vertex C ) is across side c.

Even though there are three relationships in a triangle between the three angles and their opposite sides, you only need to use two of them for the Law of Sines.

This would look like:

\cfrac{\sin(A)}{a}=\cfrac{\sin(B)}{b}

The law of sines in this form is useful to find a missing angle. By rearranging the law of sines, the reciprocal is also useful to know if you want to find a missing side length:

\cfrac{a}{\sin(A)}=\cfrac{b}{\sin(B)}\left(=\cfrac{c}{\sin(C)}\right)

What is the Law of Sines?

Common Core State Standards

How does this relate to high school math?

• High School Geometry – Similarity, Right Triangles, and Trigonometry – HSG-SRT.D.10 (+)
  Prove the Laws of Sines and Cosines and use them to solve problems.

• High School Geometry – Similarity, Right Triangles, and Trigonometry – HSG-SRT.D.11 (+)
  Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (example, surveying problems, resultant forces).

How to use the law of sines

In order to use the law of sines to find missing sides and angles:

1. Label each angle \textbf{(A, B, C)} and each side \textbf{(a, b, c)} of the triangle.
2. Use the law of sines to write the proportional relationship.
3. Solve for the missing side or angle.

Law of sines examples

Example 1: finding a missing side of a triangle

Find the length of side AB. Write your answer to two decimal places.

1. Label each angle \textbf{(A, B, C)} and each side \textbf{(a, b, c)} of the triangle.

Side AB is labeled c, so you are solving for c.

2Use the law of sines to write the proportional relationship.

Using the values known and the law of sines, the proportional relationship is:

Substitute in the values:

3Solve for the missing side or angle.

Multiply both sides of the equation by c .

Multiply both sides by 6 .

Divide both sides by \sin(55) .

Use the calculator to calculate the value of c .

Example 2: finding a missing side of a triangle

Find side BC. Write your answer to two decimal places.

Label each angle \textbf{(A, B, C)} and each side \textbf{(a, b, c)} of the triangle.

Side BC is labeled a.

Example 3: finding a missing side of a triangle

Find the length AC. Write your answer to two decimal places.

Label each angle \textbf{(A, B, C)} and each side \textbf{(a, b, c)} of the triangle.

Side AC is labeled b.

Example 4: finding a missing angle of a triangle

Find the measure of the missing angle \theta. Write your answer to one decimal place.

Label each angle \textbf{(A, B, C)} and each side \textbf{(a, b, c)} of the triangle.

Angle \theta is the same as the angle at C.

Solve for the missing side or angle.

Cross multiply to solve.

9.5\times\sin(\theta)=4.7\times\sin(86)

Divide both sides by 9.5 .

\cfrac{\sin(\theta)}{9.5}=\cfrac{4.7\sin(86)}{9.5}

\sin(\theta)=0.4935\text{ (4dp)}

Use the inverse sine function to make \theta the subject.

\sin^{-1}(\sin(\theta))=\sin^{-1}(0.4935)

\theta=\sin^{-1}(0.4935)=29.6^{\circ}\text{ (1dp)}

C=\theta=29.6^{\circ}

Be sure to check if your answer is reasonable. The larger the angle, the longer the opposite side length is. As 4.7\mathrm{~m}<9.5\mathrm{~m} , the angle at C must be smaller than the angle at B , so 29.6^{\circ} is reasonable.

Example 5: finding a missing angle of a triangle

Find the measure of the missing angle \theta. Write your answer to one decimal place.

Label each angle \textbf{(A, B, C)} and each side \textbf{(a, b, c)} of the triangle.

The angle \theta is labeled C.

Solve for the missing side or angle.

Cross multiply to solve.

\cfrac{\sin(\theta)}{7.6}=\cfrac{\sin(10)}{1.5}

1.5\times\sin(\theta)=7.6\times\sin(10)

Divide both sides of the equation by 1.5

\cfrac{\sin(\theta)}{1.5}=\cfrac{7.6\sin(10)}{1.5}

\sin(\theta)=0.8798

Use the inverse sine function to make \theta the subject.

\sin^{-1}(\sin(\theta))=\sin^{-1}(0.8798)

\theta=\sin^{-1}(0.8798)

Use a calculator to work out the value of \theta .

\theta=61.62^{\circ}

Be sure to check if your answer makes sense. As a larger side is opposite a larger angle, it’s reasonable to think that angle \theta is about 61.6^{\circ} .

Example 6: finding a missing angle of a triangle

Find the measure of the missing angle \theta.

Label each angle \textbf{(A, B, C)} and each side \textbf{(a, b, c)} of the triangle.

\theta is the angle at A.

Teaching tips for Law of Sines

• Provide opportunities for students to investigate the law of sines.

• Give students different scenarios to determine when the law of sines works. For example, when the triangle has known parts set up as ASA, AAS, SAS, SSS\text{.}

• Accurately construct triangles for students to investigate the law of sines. What happens with similar triangles? What happens if one angle is double the original size?

Easy mistakes to make

• Labeling the triangle incorrectly
  Labeling the triangle incorrectly can lead to incorrect calculations and incorrect use of the law of sines. For example, this triangle has been incorrectly labeled with the side next to the angle.
  
  
  
  Remember, the correct way is to label the angles and across from the angle is the side, for example:
  
  

• Mixing up when to use the sine and inverse sine function
  Remember that when finding a side, you have to use the sine function. When finding a missing angle, you need to use the inverse sine function (\sin^{-1}).

• Mixing up the radian measure and degree measure when using trigonometric functions
  For example, when finding a missing angle of a triangle, be sure to express the angle in degree measure. So, in your calculator be sure the ‘mode’ is in degrees, not radians.

• Using the incorrect set up for the sine rule formula
  It should not be too difficult to rearrange the formula once you have substituted the values, however small errors will make an impact on your solution.

• Canceling out sine functions
  For example, when calculating the value for a in example 2, the following calculation error is made.
  
  \begin{aligned}\cfrac{a}{\sin(41)}&=\cfrac{3.4}{\sin(27)} \\\\ a&=\cfrac{3.4\sin(41)}{\sin(27)} \\\\ &=\cfrac{3.4\times{41}}{27} \\\\ &=5.16\text{ (2dp)} \end{aligned}

Related trigonometry lessons

• Trig functions
• How to find the exact value of a trig function
• Trig formula for area of a triangle
• Trig formulas
• Trig identities
• Law of cosines
• Trig table

The next lessons are

• Circle math
• Angles of a circle
• Circle theorems

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