Askjagden's Guide to Trigonometry: Introduction to Trigonometric Functions

Askjagden's Guide to Trigonometry: Introduction to Trigonometric Functions

Functions, graphs, and relations are all quite things that you must look at to understand them. If you need more clarification, you might want to look online or in a textbook for more information on this subject. But please read this guide anyway!

Let's start with the first graph: the sine function. Using the unit circle is quite helpful for this. The sine is the y variable. The unit circle at 0 degrees is (1,0); the unit circle at 90 degrees is (0,1). The change in the y variables is 1. 90 degrees is also equal to pi/2 radians. 0 degrees is equal to 0 radians. The beginning y-coordinate is therefore 0 ((1,0)) and moves as a curve to (pi/2, 1). Then you must go to 180 degrees. 180 degrees as radians is pi. 180 degrees at the unit circle is (-1,0). The change in y is -1. Therefore the graph will subsequently go to (pi,0). Next is 270 degrees, which is (0,-1) on the unit circle. The graph will go to ((3pi/2),-1). Next is 360 degrees, which is pi/2. The graph will go to (pi/2,0). This cycle will repeat itself, both in the positive direction and the negative direction.

The domain of the sine function is obviously all of the real numbers. The range is -1 is less than or equal to y is less than or equal to 1. The range will change when transformations are made (move vertically, move horizationtally, stretch, compress, etc.). The sine function is an odd function, which means that f(-x) = -f(x) and that for every (x,y) on the graph, there is (-x,-y).

The domian of the cosine function is also all of the real numbers. The range of the cosine function is also the same as the sine function's. The cosine function is an even function, which means that f(-x) = f(x) and that the right side of the y-axis is symmetric with the left side. Here is 

Here is how the cosine function goes. Cosine is x, not y. 0 degrees starts at (1,0). 90 is at (0,1). The graph starts at (0,1). It then moves to (pi/2,0). From 90 the 180 is 0 to -1. Therefore it goes to (pi,-1). From 180 to 270, -1 goes to 0. The graph moves to ((3pi/2),0). The graph ghen goes from 0 to 1. Hence the graph moves to (2pi,1). The cycle repeats itself.

For the tangent function, there are multiple verticle asymptotes. That is because the tangent of pi/2 is undefined; it has a denominator of 0. All odd multiples of pi/2 are also vertical asymptotes. The graph starts at the origin; the graph curves up as it moves next to the asymptote on the right; it curves down as it nears the asymptote on the left. The graph repeats itself. The tangent function is the only discrete trigonometric function; the others are all continuous.

The tangent function has a range of all real numbers. Its domain is all real numbers besides pi/2 and all odd multiples of pi/2. The tangent of pi is 0, but is not undefined. The domain will change if the function is altered by transformations.

Have fun with trigonometry!