When do you use the sine rule and when the cosine rule?

The matching pair rule

In non-right triangles, regular trigonometric ratios like sine and tangent alone are not enough. Deciding between the sine rule and cosine rule comes down to whether you have a complete opposite pair.

A matching pair means you know one side and the measure of the angle facing directly across from it. If you have this matching pair plus any third piece of information, use the sine rule. The sine rule states that a / sin(A) = b / sin(B) = c / sin(C).

If you do not have a matching opposite pair, you cannot set up a simple ratio. That is when the cosine rule takes over. Specifically, use the cosine rule when you are given two sides and the angle enclosed between them (SAS), or when you are given all three sides and want to find an angle (SSS).

Step-by-step method and worked examples

First, draw the triangle and label all known measurements. Look for an angle directly opposite a known side. If you have two angles, remember that the angles in any triangle add up to 180 degrees. Subtract from 180 to find the third angle immediately rather than doing complex formulas.

Example 1 (Sine Rule): Angle A is 40 degrees, opposite side a is 10, and angle B is 65 degrees. You have a matching pair (A and a). Set up the ratio: b / sin(65) = 10 / sin(40). Multiply by sin(65) to get b = (10 * sin(65)) / sin(40), which equals roughly 14.1.

Example 2 (Cosine Rule): A triangle has side b = 7, side c = 9, and the included angle A = 60 degrees. There is no opposite pair. Use the formula: a^2 = b^2 + c^2 - 2bc*cos(A). Here, a^2 = 7^2 + 9^2 - 2(7)(9)cos(60) = 49 + 81 - 126(0.5) = 130 - 63 = 67. Taking the square root gives side a as approximately 8.19.

Triangle area and the ambiguous case

To find the area of an oblique triangle, use the formula Area = 0.5 * a * b * sin(C). The angle C must be the angle directly between sides a and b. For example, if two sides are 8 and 5 with an included angle of 30 degrees, the area is 0.5 * 8 * 5 * sin(30) = 0.5 * 40 * 0.5 = 10.

Watch out for the ambiguous case when given two sides and a non-included angle (SSA). Because sin(x) = sin(180 - x), a calculator will only return the acute angle. You must test the obtuse angle: if the obtuse angle plus the original given angle is less than 180 degrees, a second valid triangle exists, and you must calculate both.