Hamilton's Math To Build On - copyright 1993

Finding the Angles of a Right Triangle

About Math To Build On || Contents || On to Two Sides Known || Back to Ratio of Sides || Glossary

* Finding the Angles of a Right Triangle

A key to mastering the right triangles is understanding how the ratios of sides, the functions, and the angles are related. The ratios of sides are directly related to the number of degrees in the reference angle.

A relationship between the ratios of sides and the number of degrees in the angles of a right triangle has been known for years. It was found that a right triangle that has the same angles as another right triangle also has the same ratio of sides. A system was worked out which enables us to compare the ratios to a chart (these days we use a calculator) and find the degrees of the angles.

What this means is: If you take the time to learn the names of the functions and the ratios that are assigned to them, you will be able to find the degrees of the angles of any right triangle just by knowing the length of two sides.


* Using the Arc Function Keys on the Calculator

The scientific calculator has a group of keys called the arc function keys. The arc function symbol is the -1 in the upper right corner above the function name.

The arc function keys display the angle when given the correct function.

There are three arc function keys on most calculators.

The three other arc functions -- arc cosecant, arc secant, and arc cotangent -- are also used to find angles, but require another step. For the present, use the three arc function keys on the calculator.

These three functions deal with all three sides of a right triangle, and Oscar Had A Heap Of Apples will help you remember these functions.


  • First: Select the reference angle and name the legs.
    Remember : The right angle is never used as the reference angle.

  • Second: Look at the functions chart and choose the function that corresponds to the known sides of the triangle.

    Since we know the lengths of all the sides of the above triangle, any of the functions can be used.
    Let's use the sine function,

  • Third: Assign the numbers to the ratio and calculate the problem.

  • Fourth: Use the correct arc function key to find the degrees of the reference angle.

    The key is used to find the degrees of this reference angle.

    (In rounding off degrees to the nearest 1/2 degree, from 0.00 to 0.249, go down to 0. From 0.25 to .749, go to .5° . From .75 to .99, go to 1° .)

As you can see, the arc function keys react to whatever is in the display. Below, enter each number and look at the function to determine which arc function key to use, and push that key.

Finding the Angles Practice:

Find the angle for each function. Round off to the nearest 1/2° .

     (1) sin  = 0.3745          (5) tan  = 0.5785
     (2) tan  = 1               (6) sin  = 0.7756 
     (3) cos  = 0.8632          (7) cos  = 0.9397 
     (4) sin  = 1               (8) tan  1.5030 

On to Two Sides Known

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13 September 1995
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