You can use the law of tangents to solve for the measures of missing parts of a triangle when you have two sides of a triangle and the angle between them (referred to as SAS, or side-angle-side, in trigonometry). The law of tangents may look a bit intimidating, but it’s really quite nice.
In triangle ABC, with sides a, b, and c opposite the respective angles A, B, and C, the law of tangents states
![image0.png](https://www.dummies.com/wp-content/uploads/279860.image0.png)
The best way to show how the law of tangents works is with an example. In triangle ABC, a is 52, b is 28, and angle C is 80 degrees. To find the remaining parts of the triangle, follow these steps:
Use the law of tangents involving sides a and b.
Fill in the values that you know and simplify.
Multiply each side by the denominator on the right.
Determine A + B.
The sum of angles A and B equals 180 degrees minus the measure of angle C: A + B = 180 – 80 = 100.
Replace (A + B) with 100 and simplify.
Use a scientific calculator to do the calculations.
Find the value of A – B.
The difference between angle A and angle B is 39.354, or about 39 degrees.
Determine A and B by solving the system of equations for A + B and A – B.
Eliminate one of the variables by adding the two equations together.
Because A is 69.5, B equals 100 minus 69.5, or 30.5. Rounding these values to whole numbers, you get A = 70 and B = 30. (If you’re uncomfortable with rounding 69.5 up to 70, be aware that, if you had used 39.354 instead of the rounded version, 39, in the system of equations, you would have gotten a number closer to 70.)
Solve for side c by using the law of sines.
Side c is about 54 units long.