Right triangles and trigonometric functions

Consider the 3-4-5 triangle.  Because 3² + 4² = 5², this is a right triangle. Any triangle similar to this, which means that all the side lengths are increased or decreased by the same scale, is also a right triangle.

This means a 3-4-5 triangle is similar to a 6-8-10 triangle or a 30-40-50 or a .3-.4-.5 triangle. It doesn't matter if we measure these distances in inches or feet or miles or centimeters or meters or kilometers, as long as we are consistent.  In all of these triangles, corresponding angles have NOT changed.  There is the 90° angle and two angles we will call alpha and beta.  We don't currently know the measure of these two angles, but because all three angles have to add up to 180°, alpha + beta = 90°.  Any two angles that add up to 90° are called complementary.


The picture above shows us two ways to look at the right triangle, one from the perspective of the alpha angle and the other from the perspective of beta.  The hypotenuse doesn't change, but the legs are given the labels Opposite and Adjacent.  If we think of the horizontal line segment as the base and the vertical as the height, the horizontal length is Adjacent to alpha and the vertical length is Opposite alpha.

From the point of view of beta, the Opposite and Adjacent are switched, but the Hypotenuse doesn't change.

The three major trigonometric functions, sine, cosine and tangent (sin, cos, tan) can be defined as ratios between the three sides.  Remember that if the angles don't change but we change the lengths, they change by some uniform multiple, so the ratios between the side lengths will remain the same.  Here are the ratios.

Sine = Opposite/Hypotenuse
Cosine = Adjacent/Hypotenuse
Tangent = Opposite/Adjacent

The mnemonic Soh-Cah-Toa is regularly used in classes now to remember the three fractions that define the three functions.  It kind of sounds like something from a Native American language, so the story is that Soh-Cah-Toa was an important chief of some tribe, but that is completely made up.

Another important thing to remember is that if two angles are complementary, that corresponds to the letters co. 

The sine of alpha is the cosine of beta.
The cosine of alpha is the sine of beta.  

 We will later learn about three more trig functions, cotangent, secant and cosecant.  The rule about adding the prefix "co" or getting rid of it will remain the same.


The tangent of alpha is the cotangent of beta.
The cotangent of alpha is the tangent of beta.  

The secant of alpha is the cosecant of beta.
The cosecant of alpha is the secant of beta.