UNDERSTANDING RIGHT TRIANGLE TRIG
Have you ever noticed the buttons on your calculator “sin”, “cos”, and “tan”?
Those buttons are abbreviations for “sine”, “cosine”, and “tangent.”
In this activity we will look at how “sin”, “cos”, and “tan” are used in right triangles.
1. First, check that your calculator is in the right mode: "degrees".
 Be sure your that your settings are in "degrees". On the TI-83 series go to the "MODE" button and select
"Degree".
 Check your calculator. Try entering TAN and then 45. If you get "1", then you're ready to go. (Some
calculators require you to enter 45 and then TAN.)
2. Calculators know that tan 45° = 1. Let’s look closely and figure out why that is.
 Look at the following right triangles that have 45° angles.
Why do you think we call them “45-45-90” triangles?

Explain why a 45-45-90 triangle is also called an isosceles triangle.

When a calculator says that tan 45° = 1, it is telling us that two sides in our triangle have a ratio of 1.
(That is, one side is 100% of another side.)
Which two sides do you think have a ratio of 1 in our triangle?
3. Look at a right triangle with a 37° angle.
 Our calculators tell us that tan 37° = 0.754.
That means one side in the triangle is 75.4% as long as another.
Which two sides appear to be in a 0.754 ratio?

Use a ruler to verify your conjecture.
4. Look at a right triangle with a 27° angle.
 Our calculators tell us that tan 27° = 0.509. So one side in the triangle is about 50% as long as another.
Which two sides appear to be in a 0.509 ratio?

Use a ruler to verify your conjecture.
5. Let’s look at another special right triangle. The 30-60-90 triangle is really half of an equilateral triangle.
As such one side is half of another.
 Which two sides must be in a 1:2 ratio?
(That is, which side is ½ or 50% of which other side?)

Using your calculator, find which trig ratio (sin, cos, or tan) will give
you a 1:2 ratio for the 30° angle.

Using your calculator, find which trig ratio (sin, cos, or tan) will give you a 50% ratio for the 60° angle.
6. Look at a right triangle with a 15° angle.
 Our calculators tell us that cos 15° = 0.966. That means one side in the triangle is 96.6% as long as another.
Which two sides appear to be in a 0.966 ratio?

Use a ruler to verify your conjecture.
7. So we’ve seen that sin, cos, and tan all find ratios for two sides in a right triangle. Let’s see them all in action for this
next triangle. Take a look.

Our calculators tell us that sin 35° = 0.574.
Which two sides are in that ratio?

Which two sides are in the same ratio as cos 35°?

Which two sides are in the same ratio as tan 35°?
8. Try to put into your own words how sin, cos, and tan each relate to an angle in a right triangle.