When I’m teaching students various topics within trigonometry, I’m regularly struck by just how many ways there are to think about trig functions. It can be overwhelming trying to think about something new from so many different angles at the same time.
Some Trigonometry Viewpoints:
Right Triangle Trigonometry: SOH-CAH-TOA
This is how trigonometry is usually introduced. You pick a non-right angle in a right triangle, and label the sides relative to that angle: hypotenuse (the longest side, opposite the right angle), adjacent (the side touching the angle, not the hypotenuse), and opposite (the side farthest from the angle). Once the sides are labeled, students use the SOH-CAH-TOA mnemonic to make trigonometric ratios.
In case you’ve forgotten what SOH-CAH-TOA stands for:
SOH – Sine of an angle is Opposite over Hypotenuse
CAH – Cosine of an angle is Adjacent over Hypotenuse
TOA – Tangent of an angle is Opposite over Adjacent
This is often introduced right around the same time as SOH-CAH-TOA. Unfortunately, this is often before students learn about what inverse functions are (an Algebra II topic), and often before they’ve seen special triangles. Which means that inverse trig is reduced to remembering which buttons to punch in on a calculator.
In general, inverse trig functions allow you to find out what angle resulted in the ratio you have. For example, if you know the sine of a mystery angle is 0.5, then you can use inverse trig to find out that the angle has a measure of 30°.
Special Right Triangles: 30°-60°-90° and 45°-45°-90°
Why do we care about these two types of triangles in particular? Mostly because they have sides that are in ratios that we can remember, and we can derive these ratios from things we already know in case remembering doesn’t work (30°-60°-90° triangles are derived from equilateral triangles, 45°-45°-90° triangles are derived from squares). Basically these triangles let teachers ask no-calculator questions about trigonometry to find out if you understand the concepts or if you’ve mastered calculator button-pushing.
Unit Circle Trigonometry
Welcome to Algebra II, or possibly PreCalculus. This is when students find out that you can play with bigger angles than you’d find in a right triangle. It’s possible to take the Sine of 267°! You can use negative angles! It’s a whole new way of looking at trigonometry. The secret, I’ve found, is to become very familiar with the idea of reference angles (the angle formed by the terminal ray and the x-axis) and do all problems from that point of view.
This is usually the time when teachers introduce the idea of radians, an alternative way of measuring angles. As it turns out, radians are more powerful than degrees as an angle measuring system, so the more quickly you can get comfortable with them, the better (fractions are friends!).
Just when you thought your brain was exploding from thinking of triangles and circles and spinning terminal rays all at the same time, we change the context entirely with trigonometric functions. Suddenly trigonometry looks like waves rolling along the x-axis forever in both directions. I found a neat .gif explaining the connection between the unit circle and the function waves; we’ll leave the rest of the exploration of this topic for another day.
Key steps to mastering trigonometry:
- Master the art of using SOH-CAH-TOA.
- Memorize the special right triangles (this is one of very few times I’ll recommend memorization).
- Think of π radians as a half-turn, not as 180°: you want to think in radians, not waste time converting to degrees.
- You might need to spend time getting very comfortable with fractions; this is time exceptionally well spent.
- Make friends with reference angles; they’re much more powerful than you’d guess.
The more comfortable you are with each of the points above, the lower your cognitive load will be when you’re trying to solve trigonometry problems. Lowering your cognitive load will result in fewer mistakes and better comprehension of big ideas.
The other major secret to making math easier? GET MORE SLEEP!
What have you found that helps you make sense of trigonometry? Let us know in the comments! If you have a math problem you’d like to submit, click HERE.