Applying the Sum and Difference Formulas for Cosines to Find the Cosine of the Sum or Difference of Two Angles
How to Calculate an Angle Using Reciprocal Trigonometric Functions
How to Prove an Equality Using Pythagorean Identities

How to Solve a Triangle When You Know Two Angle Measures

If you know two angle measures and a side length on a triangle, you can use the Law of Sines to find the missing parts of the triangle. In this case, you need to know either two angles and the side in between them (angle-side-angle, or ASA), or two angles and a consecutive side (angle-angle-side, or AAS).

Whenever you're given two angles, you can find the third one immediately and work from there. In both of these cases, you can find exactly one solution for the triangle in question.

Solve a Triangle Using ASA

An ASA triangle means that you're given two angles and the side between them in a problem. For example, a problem could state that

image0.png

as shown in the figure. You also could be given

image1.png

This figure has all the given and unknown parts labeled for you.

A labeled ASA triangle.
A labeled ASA triangle.

To find the missing information with the Law of Sines, follow these steps:

  1. Determine the measure of the third angle.

    As a rule,

    image3.png

    So by plugging in what you know about the angles in this problem, you can solve for the missing angle:

    image4.png
  2. Set up the Law of Sines formula, filling in what you know.

    Given that the formula for the Law of Sines looks like this:

    image5.png

    the formula here sets up like this:

    image6.png
  3. Set one fraction with an unknown numerator and the fraction with a known numerator equal to each other and cross multiply.

    If you use, say, the first and third fractions, the equation look like this:

    image7.png

    Cross multiplying, you have

    image8.png
  4. Find the decimal approximation of the missing side using your calculator.

    Because sin 101 degrees is just a number, you can divide both sides of the equation by it to isolate the variable:

    image9.png
  5. Repeat Steps 3 and 4 to solve for the other missing side.

    Setting the second and third fractions equal to each other, you have this equation:

    image10.png

    This equation becomes

    image11.png

    when you cross multiply. Isolate the variable and solve for it:

    image12.png
  6. State all the parts of the triangle as your final answer.

    Some answers may be approximate, so make sure you maintain the proper signs:

    image13.png

Solve a Triangle Using AAS

In many trig problems, you're given two angles and a side that isn't between them. This type of problem is called an AAS problem. For example, you may be given

image14.png

as shown in this figure.

A labeled AAS triangle.
A labeled AAS triangle.

Notice that if you start at side b and move counterclockwise around the triangle, you come to

image16.png

This check is a good way to verify whether a triangle is an example of AAS.

After you find the third angle, an AAS problem just becomes a special case of ASA. Here are the steps to solve:

  1. Determine the measure of the third angle.

    You can say that

    image17.png
  2. Set up the Law of Sines formula, filling in what you know.

    image18.png
  3. Set one fraction with an unknown numerator and the fraction with a known numerator equal to each other and then cross multiply.

    Say that you choose to use a and b:

    image19.png

    Cross multiplying, you have

    image20.png
  4. Solve for the missing side.

    You divide by sin 68 degrees, so

    image21.png
  5. Repeat Steps 3 and 4 to solve for the other missing side.

    Setting b and c equal to each other, you have this equation:

    image22.png

    Cross multiply:

    image23.png

    Divide by sin 68 degrees to isolate the variable and solve:

    image24.png
  6. State all the parts of the triangle as your final answer.

    Your final answer sets up as follows:

    image25.png
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