Solving right triangles is possible using trigonometry so long as two sides or two angles and a side are known. The right angle is going to be 90 degrees, so there’s always at least one angle that is already known. When a person needs to determine all the sides and angles of the triangle, there are a few formulas they can use depending on the information in the problem.

Knowing Two Angles and One Side, Solve the Triangle

The right angle is always 90 degrees. If another angle is known, it’s easy to find the measurement of the third angle. Simply subtract the second angle from 90 degrees to get the third angle. For instance, if the second angle is 32, the three angles for the triangle would be 90, 32 and 58.

When one side is known, it’s possible to determine the remaining sides using a specific formula. The unknown side divided by the known sign is going to be sin of the angle across from the unknown side. For example, angle A is 38 degrees and side b is 10. To find side a, the equation would look like: a/10=sin 38.

At this point, the student can use the three-place trigonometric table to determine sin 38 or a calculator. The equation is then a/10=0.616. Solve for a, which is 6.16. Now that two sides are known, it’s possible to determine the third side of the triangle using the Pythagorean theorem, a^2+b^2=c^2. Solving this equation, the three sides are 10, 6.16, and 11.75

Knowing Two Sides

When two sides are known, the third one can be found using the Pythagorean theorem as shown above. After this, it’s important to determine the angles. One of the angles is going to be 90 degrees, so only two more are needed. Assume for the following example that angle C is 90. Sides are a=10, b=6, and c=8. To find the second angle, use cos A=6/10 (the 8 and 10 are because these are the sides connected to the angle). This means cos A=0.6, but an angle is needed. Use the three-place trigonometric table to determine the angle for A, which is 37 degrees.

Once the second angle is known, the third one can be determined by subtracting the second one from 90. In this example, the second angle would be 53.

Solving for a right triangle is made easier because one of the angles is always known. So long as a second angle and a side or two sides are known, it’s possible to solve all the angles and sides. In more advanced problems, knowing an angle outside of the triangle can help a person solve all of the angles inside it as well. Practice makes it easier to remember what formulas are needed to solve for what the person needs and for the student to learn how to easily use the three-place trigonometric table. Try a few problems today to get a little more practice solving all parts of these triangles.