Hamilton's Math To Build On - copyright 1993

The Complementary Angle

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* Complementary Angle

Notice the 53° angle on the far right side of the same row. This is the other angle in the 3-4-5 triangle and is called the complementary angle. A complementary angle is found by subtracting the reference angle from 90° (90° - 37° = 53° ).

Since the reference and complementary angles are in the same triangle, both angles use the same numbers (length of sides) to derive their functions. The sine of the reference angle is equal to the cosine of the complementary angle, and vice versa.

The following charts show which functions of the reference and the complementary angles are equal to each other.

Notice that any two functions
opposite each other are
identified by a name and
its coname.

Note: If you name
the sides incorrectly, you will
end up with the complementary
angle instead of the reference angle.
One way to avoid this error is to
remember that the shortest side
is always opposite the smallest

To solve any math problem, it is necessary to know a certain amount of information. The information given in this book, referred to as knowns, is enough information to begin solving the problem presented; however, you may need to add to those knowns by further calculations to arrive at the final answer. The needs are the information you are trying to find.

Here is a problem which uses the table to find the angles.

Find the angles of this triangle.

  • First: Draw a thumbnail sketch, then label the reference angle and name the sides.

      Adjacent side = 16'

      Opposite side = 8'


  • Second: Choose the function that uses
    the knowns to determine the needs.
    Replace the ratio with the knowns and
    calculate the problem.

    There are two functions which can be
    used when you know just the opposite
    and adjacent sides: tangent or cotangent.

    Cotangent is used this time.

  • Third: Refer to the function table to determine the angle.

    Look at the functions table on pages 214 and 215 under cot for 2. The closest angle is 26.5° , or 26 1/2° .

    The complementary angle is 90° - 26.5° = 63.5° , or 63 1/2° . Did you notice the complementary angle on the right side of the table?

Finding the Angle Using the Functions Table: Practice

Find the unknown angles for each right triangle below. Use the functions table.


On to Using the Function Keys

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16 September 1995
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