A FEW MEMORY AIDS FOR BASIC TRIGONOMETRY
TABLE OF CONTENTS
If you know algebra, basic trig's a cinch, given the tricks I present below. Let's begin.
BASIC RIGHT TRIANGLE TRIG FUNCTIONS
Say the following out loud: SOH (so),CAH (cah), TOA (toeah). It should sound a little like "soak a toe (ah)". Remember this.
What does this stand for? The basic trig functions, sine (Sin), cosine (Cos), and tangent (Tan), of course:
S in Ø = O pposite side length/ H ypotenuse length C os Ø = A djacent side length/ H ypotenuse length T an Ø = O pposite side length/ A djacent side 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.
Return to Top
WHEEL TRIG FUNCTIONS
Take a look at the "wheel" above. Burn the picture of the wheel into your mind. Note that the TanCot line divides the wheel in NorthSouth 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 WestEast 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.
Return to Top
THE RECIPROCAL TRIG FUNCTIONS
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,
Cot Ø = 1/(Tan Ø)





Csc Ø = 1/(Sin Ø)




Sec Ø = 1/(Cos Ø)


That gets us three more trig functions. There are more to be found on the wheel.
Return to Top
OTHER TRIG FUNCTIONS ON THE WHEEL (Product and Quotient Relationships)
Now look at the functions immediately adjacent to Tan, it's true that:
Tan Ø = Sin Ø Sec Ø





Looking at the functions immediately adjacent to Sin, it's also true that:




Sin Ø = Cos Ø Tan Ø




The following is also true:


Cos Ø = Cot Ø Sin Ø




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.
Now look at the functions on the side of the circle to the left of Tan, the following is true:





Tan Ø = Sin Ø / Cos Ø




A similar relationship exists with the functions on the side of the circle to the right of Tan:




Tan Ø = Sec Ø / Csc Ø


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.
Return to Top
ANGLE SUM AND DIFFERENCE FORMULAS
Say this aloud: sinecosine, cosinesine, cosinecosine, sinesine. Remember what this sounds like; note the almost singsong rhythm.
What does it stand for? The angle sum and difference trig formulas:
Sin (a + b) = Sina Cosb + Cosa Sinb Sin (a  b) = Sin a Cos b  Cos a Sin b Cos (a + b) = Cosa Cosb  Sina Sinb Cos (a  b) = Cos a Cos b + Sin a Sin b
Return to Top
SUMS OF THE SQUARES OF TWO FUNCTIONS
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:
Sin^{2}Ø + Cos^{2}Ø = 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 canthrough algebraic manipulationderive most other trig formulas you might need.
Return to Top
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 trigonometryall 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.
. Return to Top
Back to the Math/Science Main Page
