S in Ø = O pposite side length/ H ypotenuse length
Burn the trangle and the definitions into your mind. We now know how to compute the three main trig functions. Next we build upon that by looking at "Wheel Functions", the inverse, product, and quotient trig functions.
Take a look at the "wheel" above. Burn the picture of the wheel into your mind. Note that the Tan-Cot line divides the wheel in North-South fashion, that the functions beginning in the letter S are at the top of the wheel in a West- East orientation, and that those functions beginning with the letter C are at the bottom of the wheel in an West-East orientation.
Now pick any trig function on the wheel, for instance, the tangent (Tan). The positions of the other trig functions on the wheel tell you their relationship to Tan.
Notice that the spokes of the wheel each join only two of the trig functions named. The trig functions along the same spoke are the reciprocals of one another. In other words,
That gets us three more trig functions. There are more to be found on the wheel.
Now look at the functions immediately adjacent to Tan, it's true that:
and so on, moving around the wheel.
The rule to remember is that the function equals the product of its two immediately flanking functions (multiply 'em). These are the product relationships.
Pick any function on the wheel, and similar relationships exist with the pairs of functions on either side of it. So the rule to remember is: the function equals the quotient of the two functions (divide 'em) that lie on the same side of the circle between the function and its inverse. These are the quotient relationships.
Say this aloud: sine-cosine, cosine-sine, cosine-cosine, sine-sine. Remember what this sounds like; note the almost sing-song rhythm.
What does it stand for? The angle sum and difference trig formulas:
Sin (a + b) = Sina Cosb + Cosa Sinb
Cos (a + b) = Cosa Cosb - Sina Sinb
Look at the "Unit Circle" (a circle with a radius of 1) below:
Note the right triangle within the circle. The hypotenuse of that triangle is 1 ; the side opposite the angle Ø is sin Ø (i.e., sin Ø = o pposite side/hypotenuse; and since the hypotenuse is 1, sine Ø = opposite side). The adjacent to angle Ø is cos Ø. So, as a result of the Pythagorean Theorem which says that the sum of the square of the hypotenuse is equal to the sums of the squares of the two sides:
Sin2Ø + Cos2Ø = 1
Now that you have all the basic trig functions, all the wheel trig relationships, the angle sum and difference formulas, and the unit circle squares of functions formula, you can--through algebraic manipulation--derive most other trig formulas you might need.
Historical note: This approach to trig is based on the unique way my St. Ignatius High School (Cleve.,OH) geometry teacher, Frank Bitzan, taught us trigonometry--all in two weeks. After almost 40 years, I haven't forgotten the trig, so I guess his approach worked. I hope it does for you too.