basic trig in one easy lesson

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A FEW MEMORY AIDS FOR BASIC TRIGONOMETRY

TABLE OF CONTENTS

Wheel Trig Functions

Reciprocal Trig FunctionsProduct & Quotient Trig Relationships

Angle Sum and Difference Formulas

Sums of Squares Functions

Historical Note

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 (c-ah), TOA (toe-ah). 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 lengthT 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.

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WHEEL TRIG FUNCTIONS

Trig in One Easy Lesson http://www.therthdimension.org/MathScience/trigtricks/body_trig

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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 bottomof 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.

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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.

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OTHER TRIG FUNCTIONS ON THE WHEEL (Product and Quotient Relationships)

Now look at the functions immediately adjacent to Tan, it's true that:


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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 leftof 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.

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ANGLE SUM AND DIFFERENCE FORMULAS

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

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

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SUMS OF THE SQUARES OF TWO FUNCTIONS

Look at the "Unit Circle" (a circle with a radius of 1) below:


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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 functionsformula, you can--through algebraic manipulation--derive most other tr ig formulas you might need.

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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.

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3/9/2011

basic trig in one easy lesson

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