In precalc 11 this week, we learned about trigonometry, and more specifically, special triangles in which you don’t need a calculator. There are two types of triangles we learned to memorize, which look like this:

As you can see, these triangles involve the angles 45°, 30°, and 60°. They also have the three primary trigonometric ratios: sine, cosine, and tangent. For the second triangle, the ratios can be based off both the 30° or 60°. Here are the ratios:

Below are some questions where the angle is given, and you need to find the ratio (or fraction) for either sine, cosine, or tangent. You can start by drawing a diagram of the four quadrants, labelling each with sine, cosine, tangent, or all. This tells you in what quadrant each of the ratios will be positive. Then figure out where the angle is from the question, and use it to draw the triangle. By doing this, you can find the reference angle, and therefore choose which of the special triangles to use.

Next, refer back to the original ratios stated above, and find the one to fit the reference angle. If the sign (positive or negative) of the angle is in a quadrant that matches the sign, keep it positive. If not, change the answer to negative.

Remember, these questions can all be done without a calculator if you remember the two special triangles. As long as either 45°, 30°, or 60°, is present, they are easy to solve.