How can I use the Law of Sines to find the missing side ‘b’ of a triangle?

by

The Law of Sines is a useful formula in trigonometry that relates the lengths of the sides of a triangle to the sines of its angles. To find the missing side ‘b’ of a triangle using the Law of Sines, you need some information about the triangle, such as the lengths of one or two sides and the measures of one or two angles.

The Law of Sines states:

a/sin(A) = b/sin(B) = c/sin(C)

Where:

  • a, b, and c are the lengths of the sides of the triangle.
  • A, B, and C are the angles opposite those sides.

To find side ‘b’, you can rearrange the Law of Sines formula like this:

b = a * (sin(B) / sin(A))

Here’s a step-by-step method you can follow:

  1. Identify the known values:

    • Side ‘a’ (a known length)
    • Angle ‘A’ (the angle opposite side ‘a’)
    • Angle ‘B’ (the angle opposite side ‘b’, which you want to find)

  2. Make sure you have all necessary angles or sides:

    If you’re given two angles and one side, you can find the third angle using the fact that the sum of the angles in a triangle is always 180 degrees. Once you have angle ‘B’, you can proceed.

  3. Plug the values into the formula:

    Substitute in your known values into the rearranged formula:

    b = a * (sin(B) / sin(A))

    Calculate the value of side ‘b’.

  4. Check your work:

    Make sure the value makes sense in the context of your triangle, considering the properties and relationships of its angles and sides.

Here’s an example for clarity:

Suppose you know:

  • Side ‘a’ = 10 units
  • Angle ‘A’ = 30 degrees
  • Angle ‘B’ = 45 degrees

First, find angle ‘C’:

C = 180 - A - B = 180 - 30 - 45 = 105 degrees

Now, plug the values into the Law of Sines:

b = 10 * (sin(45) / sin(30))

This will give you the length of side ‘b’. Calculate the sines:

  • sin(45) ≈ 0.7071
  • sin(30) = 0.5

Substituting those into the equation:

b ≈ 10 * (0.7071 / 0.5) = 10 * 1.4142 = 14.142 units

Therefore, in this scenario, side ‘b’ would be approximately 14.142 units long. Using the Law of Sines effectively enables you to find missing sides quite systematically.

Leave a Comment