This week in math we learned about so called “special triangles” which are triangles that have reference angles of 60, 30, and 45. This understanding is important to triganomatry because it sets the foundation for trig in the future, and encorporates other things we have learned in math in the past. I chose “special triangles” for this blog post because I find that once you have identfied the triangles as “special”, the math is quite straight forward and manageable.
Lets look at some examples…
Step 1. The first step is to get a visual of your angle by drawing it on your x, y axis.
Step 2. Now we want to identify our principle angle, to do so, I add 360 (full rotation) as many times as needed until I reach a positive value, which happens to be 120.
Step 3. Now that we have identfied our principle angle, lets identify our reference angle. I subtract 120 from 180 to get 60. Since 60 degrees is one of our special angles, we do not need a calculator for this problem.
Step 4. Draw the triangle and label its sides.
Step 5. Determine the exact values of 3 primary trig ratios. Recall the formulas for sin, tan, and cos. Since the instructions call for exact, leave the answer as a fraction.
Example 2.
Step 1. Draw angle on standard x, y axis.
Step 2. Find principle angle – I subtract 360 from 1305 until I get my principle angle of 225 degrees.
Step 3. Find reference angle. I subtract 180 from 225 and get my reference angle of 45 degrees.
Step 4. Since 45 degrees is a special triangle, draw out and label all sides of the special triangle for 45 degrees.
Step 5. Determine the exact values of 3 primary trig ratios. Recall the formulas for sin, tan, and cos. Since the instructions call for exact, leave the answer as a fraction.
Glossary :
Reference angle : the positive angle between the terminal side of θ and the x-axis.
Principle angle – the counter-clockwise angle between the initial arm and the terminal arm of an angle in standard position.