Sine Law

Thus far, we’ve learned how to efficiently solve right triangles using trigonometric functions. Remember that to “solve” a triangle means we’re finding all its unknown lengths and angles. What if we have a triangle that is not a right-angle triangle (called an oblique triangle)?

If your oblique triangle has the following configurations:

  • Angle – Angle – Side (AAS)
  • Angle – Side – Angle (ASA)
  • Side – Side – Angle (SSA)

… the sine law is used (shown below), and it doesn’t matter whether your oblique triangle is acute or obtuse. The sine law formula is shown below:


Where the capital letters (A, B, and C) represent the angles, and the small letters (a, b, and c) represent the length of the opposite side relative to the angle.

Keep in mind from previous sections that your calculator will always give an acute angle between 0° to 90° with positive ratios. However, we know sine is positive in both the 1st and 2nd quadrant, so sometimes while solving for an angle using the sine law, it might get the wrong angle after solving – so be careful. You will have to decide whether to use the angle you obtain from your calculator or subtract it from 180° to get the obtuse version instead. All of this will make more sense to you when we look at the “ambiguous case” example later.

Let’s watch two examples where the sine law applies:

As mentioned earlier, sometimes you’ll have to decide whether the angle you solved for is the correct one according to the diagram. In other words, the angle illustrated might look bigger than the angle you find. For example, you find that the angle 65°, yet it looks more like 115°. In that case, you subtract your acute angle from 180° to get the obtuse version instead. This is known as an ambiguous case and an example is provided below. In case you’re not given a diagram, this can only happens with obtuse triangles.

For good measure, see if you can try answering this next question on your own.

A 71.6-m-high antenna mast is to be placed on sloping ground, with the cables making an angle 42.5° of with the top of the mast. Find the length of each cable.