Use the Sine Rule

The Sine Rule, also known as the law of sines, is exceptionally helpful when it comes to investigating the properties of a triangle. While the three trigonometric ratios, sine, cosine and tangent, can help you a lot with right angled triangles, the Sine Rule will even work for scalene triangles. Regardless of the shape of the triangle, if you know some limited information about its angles and sides, you can use the Sine Rule to calculate the rest.

Steps

Labelling the Triangle

  1. Mark the sides. The sides of a triangle are traditionally marked with three consecutive letters, usually A, B and C. The order that you choose to mark the sides generally does not matter, unless something in the problem you are working on specifies it.[1]
  2. Mark the angles. Mark the three angles of the triangle with letters that correspond to the side lengths. For example, if you use capital letters A, B and C for the sides, then mark the angles with lower case letters a, b and c. You can also use lower case Greek letters <math>\alpha, \beta, \text{and }\gamma</math>. Place these so they correspond with the labeled sides, so angle <math>\alpha</math> is opposite side A, angle <math>\beta</math> is opposite side B, and angle <math>\gamma</math> is opposite side C.[1]
    • One way to determine that a side is “opposite” a chosen angle is to make sure that it does not form one of the rays of the angle. If labeled correctly, angle <math>\alpha</math> wll be formed by the two sides B and C. It will therefore be “opposite” side A.
    • Similarly, angle <math>\beta</math> is formed by sides A and C and is opposite side B.
    • Angle <math>\gamma</math> is formed by sides A and B and is opposite side C.
    • Some math texts will use capital letters for the sides and lower case for the angles. Others do the opposite. It does not matter, as long as you are consistent.
  3. Label any measurements that you know. In your problem, you must be given some side and angle measurements. You should mark these on your sketch of the triangle.[1]
    • You may be able to calculate one or more measurements using some rules of geometry.
      • For example, if you are told that the triangle is isosceles, then you are able to mark that two of the angles are equal, as well as the two corresponding sides.
      • As another example, if you are told that two angles are 40 and 75 degrees, you can then calculate the third angle to be 65 degrees, since all three angles must add up to 180 degrees.

Calculating with the Sine Rule

  1. Understand the Sine Rule. The Sine Rule, also called the law of sines, is a rule of trigonometry that relates the sides of a triangle and its angle measurements. While most of trigonometry is based on the relationships of right triangles, the law of sines can apply to any triangle, whether or not it has a right angle.[2]'
    • The law of sines is stated as follows:
      • <math>\frac{A}{\sin \alpha}=\frac{B}{\sin \beta}=\frac{C}{\sin \gamma}</math>
    • The same rule can be rearranged to yield the following equivalent statements:
      • <math>\frac{\sin \alpha}{A}=\frac{\sin \beta}{B}=\frac{\sin \gamma}{C}</math>
  2. Review the data you need. For the law of sines to be useful, you must know the measurements of at least two angles and one side, or two sides and one angle. In either case, you must have at least one pair that consists of a side and its opposite angle.[2]
    • For example, the following combinations would be sufficient for the law of sines to apply:
      • Side A, Side B and angle <math>\alpha</math>
      • Side A, Side C, and angle <math>\gamma</math>
      • Side B, angle <math>\beta</math> and angle <math>\alpha</math>
    • The following combinations are examples that would NOT be sufficient to apply the law of sines:
      • Side A, Side B and Side C. (This does not work because you have no angle measurement.)
      • Side A, Side B and angle <math>\gamma</math>. (This does not work because the known angle is not opposite either of the known sides.
      • Side B, angle <math>\alpha</math> and angle <math>\gamma</math>. (This does not work because the known side is not opposite either of the known angles.)
  3. Write the portion of the law of sines that you need. The law of sines works to help you find one piece of information about a triangle -- a side or an angle measurement -- if you know three others. While the full law of sines is written as a three-part equation, you only need to equate two for the rule to work.[2]
    • For example, if you know sides A and B and angle <math>\alpha</math>, then you need the portion of the law of sines that says:
      • <math>\frac{A}{\sin \alpha}=\frac{B}{\sin \beta}</math>
    • Notice the similarity of the law. It really doesn’t matter which label you use for any sides or angles. The important thing to remember is that you are comparing ratios. The ratio of any side to its opposing angle is equal to the ratio of any other side to its opposing angle.
  4. Fill in the numbers that you know. Suppose you are given that side A is 12, angle <math>\alpha</math> is 80 degrees, and angle <math>\beta</math> is 40 degrees. Find the length of side B. You can mark these numbers on the triangle and set up the problem as follows:[2]
    • <math>\frac{A}{\sin \alpha}=\frac{B}{\sin \beta}</math>
    • <math>\frac{12}{\sin 80}=\frac{B}{\sin 40}</math>
  5. Rearrange to solve for the unknown information. Use basic algebra to maneuver the unknown information to stand alone on either side of the equation. You can then reduce the problem to find the answer.[2]
    • <math>\frac{12}{\sin 80}=\frac{B}{\sin 40}</math>
    • <math>\frac{12\sin 40}{\sin 80}=B</math>
    • <math>7.83=B</math>
    • To find the value of the sine of an angle, such as <math>\sin 40</math> in the problem above, you can use most handheld calculators with trigonometric functions. Different calculators operate differently. With some calculators, you will enter your angle measurement first and then the "sin" button. With others, you will enter the "sin" button first and then the angle measurement. You will have to experiment with your calculator.
    • Alternatively, there are some tables available either in math books or online. With a trigonometry table, you can find your desired angle measure in one column and the corresponding value of sine, cosine or tangent in another column.

Practicing with Other Problems

  1. Solve for an unknown angle. Suppose, as a different problem, that you know two sides and need to solve an unknown angle. You are given that side A is 10 inches long, side B is 7 inches long, and angle <math>\alpha</math> is 50 degrees. You can use this information to find the measurement of angle <math>\beta</math>. Set up the problem as follows:[2]
    • <math>\frac{A}{\sin \alpha}=\frac{B}{\sin \beta}</math>
    • <math>\frac{10}{\sin 50}=\frac{7}{\sin \beta}</math>
    • <math>\sin \beta=\frac{7 \sin 50}{10}</math>
    • <math>\sin \beta=\frac{7*0.766}{10}</math>
    • <math>\sin \beta=0.536</math>
  2. Use the inverse function if needed to find the angle. In the above example, the law of sines provides the sine of the selected angle as its solution. To find the measure of the angle itself, you must use the inverse sine function. This is also called the arcsine. On a calculator, this is generally marked as <math>\sin^{-1}</math>. Use this to find the measure of the angle.[3]
    • For the example above, the final step is as follows:
      • <math>\sin \beta=0.536</math>
      • <math>\beta=\arcsin 0.536</math>
      • <math>\beta=32.4</math>.
  3. Solve a problem with incomplete information. Suppose you are told that angle <math>\alpha=30\text{ degrees}</math>, angle <math>\beta=50\text{ degrees}</math>, and side C, which connects them, is 10 inches long. Find the measurement of all sides and angles for the triangle.
    • First, you should recognize that you do not yet have enough information for the sine rule to apply. The sine rule requires that you have at least one pair with an angle that opposes a known side. However, you can calculate the third angle of this triangle using simple subtraction. All three angles add up to 180 degrees, so you can find angle <math>\gamma</math> by subtracting:
      • <math>\gamma=180-\alpha-\beta=180-30-50=100</math>
    • Now that you know all three angles, you can use the sine rule to find the two remaining sides. Solve them one at a time:
      • <math>\frac{C}{\sin \gamma}=\frac{B}{\sin \beta}</math>
      • <math>\frac{10}{\sin 100}=\frac{B}{\sin 50}</math>
      • <math>\frac{10 \sin 50}{\sin 100}=B</math>
      • <math>\frac{10*0.766}{0.985}=B</math>
      • <math>7.78=B</math>
    • Thus, side B is 7.78 inches long. Now solve for the final remaining side.
      • <math>\frac{C}{\sin \gamma}=\frac{A}{\sin \alpha}</math>
      • <math>\frac{10}{\sin 100}=\frac{A}{\sin 30}</math>
      • <math>\frac{10 \sin 30}{\sin 100}=A</math>
      • <math>\frac{10*0.5}{0.985}=A</math>
      • <math>5.08=A</math>
    • Side A, therefore, is 5.08 inches long. You now have all three angles, 30, 50 and 100 degrees, and all three sides, 5.08, 7.78, and 10 inches.

Tips

  • Notice that if you have the two sides and an angle, or two angles and a side, that you need to use the law of sines, you can then use the law of sines repeatedly to find all the remaining angle and side measurements of the triangle. Once you know two angles, you can find the third by subtracting from 180 degrees. Then, with the third angle, you can repeat the law of sines to find the third side length.
  • In addition to the law of sines, you should also learn the law of cosines. The law of cosines is a different arrangement of sides and angles that can also help you learn information about a triangle.


Related Articles

Sources and Citations

  1. ↑ 1.0 1.1 1.2 https://www.mathsisfun.com/algebra/trig-sine-law.html
  2. ↑ 2.0 2.1 2.2 2.3 2.4 2.5 http://www.mathwarehouse.com/trigonometry/law-of-sines/formula-and-practice-problems.php
  3. ↑ http://www.rapidtables.com/math/trigonometry/arcsin.htm#definition
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