You first must determine which ratio you need to use in reference to the given reference
angle of 45⁰. Since you're given the length of the side opposite and you're finding the
hypotenuse length, you need to use the sine ratio.

sine = opposite side                     sin 45⁰ = 5.715
            hypotenuse                             1                x

Once you set up the ratio, you then find the sin 45⁰ on your calculator. It is .7071.
You now replace this value in the ratio you have set up.

sin 45⁰ = 5.715   becomes     .7071 = 5.715
     1              x                                  1             x

                    To solve for x, you multiply across the equal sign.

                                        .7071(x) = 5.715(1)

                                        .7071x = 5.715

                    Now divide each side by .7071.

                                        .7071x = 5.715
                                         .7071       .7071

                                            x = 8.0823…ft.

                    Rounding to the nearest hundredth, x = 8.08 ft.

            Thus trigonometric functions are used to find missing measurements when
you only know the information of a reference angle and one side of the triangle.

What happens if the same triangle was given, only this time, you were
asked to determine the length of the other leg and not the hypotenuse?
The procedure would be slightly different because you could not use the
sin ratio. Which ratio would you need? If you know the side opposite the
reference angle and you're looking for the side adjacent to the reference
angle, which ratio would you use? You'd use the tangent ratio of course.

So now the problem goes like this.

Example 7:

5.715 ft.

?

tan 45⁰ = 5.715 This time we use the tangent ratio.

1. x

1.0000x = 5.715 Multiply across the equal sign.

Example 8:

x

cos 60⁰ = . x Set up the ratio.

1. 25.15

.5000 = . x . Find the cos 60⁰ on your calculator.

1. 25.15