At this point, we'll look back and briefly review what we can now do with angles, trigonometric functions, and expressions.
There are a number of different things we can do with angles at this point.
- We can convert between different units of measure.
- Given an angle in standard position, we can calculate which quadrant it lies in, what its corresponding reference angle is, and the length of the arc it intercepts on the unit circle.
- We also know the sign of the trigonometric functions of a given angle.
Functions, Trigonometric Functions, and Graphs
Drawing and understanding graphs is important for trigonometry. By now we should be familiar with the graphs for the six trigonometric functions, as well as the variations of those graphs. We also know how to test whether a given graph is a function, using the vertical line test.
Also, using the eight fundamental identities and the negative angle identities, we can simplify trigonometric equations and create new identities.
First Steps Toward Future Trigonometric Greatness
Before we continue our study of more complex trigonometry, we should stop and formally learn a few of the values of the trigonometric functions for the most basic angles.
So far we can evaluate trigonometric functions only when we know a point on the terminal side of an angle in standard position. With graphs, we can estimate the value of a function at a given point. It is often necessary, though, to evaluate a trigonometric function at a specific angle, knowing only the angle's measure. Generally speaking, to evaluate a trigonometric function at a specific angle, we must use a calculator. But for a few angles, the ratios created by the sides of the angles are not complex, and the values of the trigonometric functions can be easily memorized. For the following angles, θ, measured in radians, the ratio of the x-coordinate to the y-coordinate to the distance from the origin (x : y : d ) is this:
- θ = ; :1:2 .
- θ = ; 1:1:.
- θ = ; 1::2.
These ratios, as you may have recognized, come from the ratios of the sides of certain special triangles. It will be useful to memorize these ratios, so that you can calculate the values of the trigonometric functions at these angles in your head. Using these ratios, reference angles, and knowledge of the signs of the functions in different quadrants, it is possible to evaluate trigonometric functions are many common angles without using a calculator.