Hamilton's Math To Build On - copyright 1993

Using One Side and One Angle

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* Calculating Right Triangles Using One Side and One Angle

So far we have used the functions only to determine angles of the right triangle. We are now going to use those same functions to find the unknown sides of the right triangle when one side and one angle (besides the right angle) are known.

A method for determining leg lengths can be found by rearranging the functions and the ratios. You can multiply both sides of a formula by the same number or function and the formula will remain equal.

For example,

If you multiply both sides of the cosecant ratio by the opposite side (denominator of the ratio), you can determine a formula to find the length of another side.

The two opposites on the right side of the equal mark cancel each other out and leave us with this formula:

Opposite side x Cosecant = Hypotenuse

This formula indicates that the length of the opposite side (of a right triangle) times the cosecant (of the reference angle) equals the length of the hypotenuse.

All of the functions can be rearranged using this method.

To make the next chart easier to read, the terms have been placed in a different order. This chart will be referred to as the NAK chart.

The NAK chart shows which functions (of the 
angle) should be multiplied by the known side to
get the needed side.

Notice how each side of a right triangle can be found
if either of the other sides and one angle (other than the
90° ) are known.

Remember: Use the NAK chart if you know one angle
(other than the right angle) and the length of one side.


A right triangle has a hypotenuse with a length of 4'
and an angle of 35° , and you want to determine the
length of the other sides.

  • First: Determine your knowns and needs.

    The knowns are the hypotenuse of 4" and the reference angle of 35° .
    The needs are the opposite and adjacent sides.

  • Second: Use the NAK chart to determine which functions use the known side
    to find the needed sides.

Notice that there is one function for each needed side that uses the known side.

    Let's first find the opposite side:

    The length of the side opposite the reference is 2.2943', or 2' 3 9/16".

To find the length of the adjacent side when the length of the hypotenuse and an angle are known, use the cosine of the angle.

    The length of the adjacent side is 3.277', or 3' 3 5/16".

Calculating Right Triangles Using One Side and One Angle: Practice

Find the lengths of the two unknown sides (the angle given is the reference angle for the side). Round off your answers to 4 decimal places.

          Deg    Side   Lg

       (1)  65°      opp     4"
       (2)  45°     hypo     5'
       (3)  20°      adj     12"
       (4)   1°      opp     1" 
       (5)  37°      adj     6'
       (6)  55°      opp     3'
       (7)  70°      hyp     136"
       (8)  88°      hyp     99'

The functions may be found with the calculator
or thefunctions tables.


Note: You now have two ways of finding the sides of a right triangle.

  1. If you know two sides, you can use the Pythagorean Theorem.

  2. If you know one angle and the length of one side, you can rearrange the functions and ratios or use the NAK chart.

On to Seeing is Believing

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