In trigonometry, there are 6 trigonometric functions that relate an angle of a right triangle to two side lengths. The 3 main functions are sine, cosine, and tangent, with their reciprocals being the cosecant, secant, and cotangent.
Let’s remember the Pythagorean Theorem:
a^2+b^2=c^2 \\ c=\sqrt{a^2+b^2}We can visualize the Pythagorean theorem in all four quadrants and the way that it affects the x, y, and r values (and whether they are positive or negative — this is important!).
These are the four quadrants — memorize them!
In Quadrant I, the x value, y value, and r value are always positive (the r value being the diagonal line that connects x and y and makes a triangle).
r^2=x^2+y^2 \\ r=\sqrt{x^2+y^2}In Quadrant II, the x value is negative while the r and y values are always positive.
In Quadrant III, both x and y are negative, and the r is positive.
In Quadrant IV, x and r are positive, and y is negative.
The Six Trig Functions
sin\theta=\frac{y}{r} \ \ \ cos\theta=\frac{x}{r} \ \ \ tan\theta=\frac{y}{x}csc\theta=\frac{r}{y} \ \ \ sec\theta=\frac{r}{x} \ \ \ cot\theta=\frac{x}{y}As we can see, we have three functions (sine, cosine, and tangent) with three inverse functions (cosecant, secant, and cotangent). Since and cosecant are inverse, cosine and secant are inverse, and tangent and cotangent are inverse.
Quadrantal Angles
Quadrantal angles are angles that terminate on the y-axis or the x-axis. Each of these quadrantal angles fall into one of the four quadrants.
The angles are as follows:
0\degree,90\degree,180\degree,270\degree,360\degree
0 degrees is quadrant 1, 90 degrees is quadrant 2, 180 degrees is quadrant 3, and 360 degrees is quadrant 4.
Recall the Pythagorean theorem presented earlier.
So let’s say that we have an example such as this:
Angle theta has the point (-6, -8) on the terminal side. Find the 6 trig functions. Well, we know that x is equal to -6 and y is equal to -8. Let’s find r.
r = \sqrt{x^2 + y^2} \ \ = \sqrt{-6^2 + -8^2} \ \ = \sqrt{36 + 64} \ \ = \sqrt{100} \ \ = 10We now see that our r is equal to 10, and we can use this information to fill in all the 6 trig functions.
sin\theta = \frac{y}{r} = \frac{-8}{10} = \frac{-4}{5} cos\theta = \frac{x}{r} = \frac{-6}{10} = \frac{-3}{5}tan\theta = \frac{y}{x} = \frac{-8}{-6} = \frac{4}{3}csc\theta = \frac{r}{y} = \frac{10}{-8} = \frac{5}{-4}sec\theta = \frac{r}{x} = \frac{10}{-6} = \frac{5}{-3}cot\theta = \frac{x}{y} = \frac{-6}{-8} = \frac{3}{4}Working out these functions makes it easy to see the how these functions are put together and how their inverse relationships work. Once we find the first three functions, we can simply invert them to find the other three and save ourselves a little bit of time!
***TIP: If any of your functions have a number over 0, then that function is simply “undefined”.