# Use the Law of Tangents with SAS

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

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.