THE SIX TRIGONOMETRIC RATIOS

Copyright by Ingrid Stewart, Ph.D. Please Send Questions and Comments to ingrid.stewart@csn.edu.

Learning Objectives - This is what you must know after studying the lecture and doing the practice problems!

1. Define a right triangle and a unit circle.

2. Memorize and use the six trigonometric ratios in a right triangle..

3. Memorize and use the reciprocal and quotient identities

We will now continue our study of trigonometry by defining six trigonometric ratios. They are fundamental to almost everything else we do in this course.

The six trigonometric ratios can be defined by using a

right triangleor theunit circle. In this lecture, we will use the right triangle approach. Let's quickly review both.

Review - The Right TriangleA triangle in which one angle is

is called a right triangle. The side opposite the right angle is called the90^{o}hypotenuseand the remaining two sides are calledlegs.In the picture below, the length of the hypotenuse is

and the lengths of the two legs arecanda.bThe sides of a right triangle are related via the

Pythagorean Theoremas follows:

Review - The Unit CircleThe

unit circleis a circle of radius 1.The equation of the

unit circlein a rectangular coordinate system with center at the origin is .

Definition of the Six Trigonometric Ratios - see #1 through 4in the "Examples" documentTrigonometric ratios are always based on some angle.

We will use angle in a right triangle to define the six trigonometric ratios.In the definitions of the trigonometric ratios, we will use the following abbreviation:

adj- the side of a right triangleadjacent to the angle

opp- the side of a right triangleopposite the angle

hyp- thehypotenuseof a right triagle

Sine Ratio:Pronounce

as "sine theta".sin

Cosine Ratio:Pronounce

cosas "cosine theta".

Tangent Ratio:Pronounce

tanas "tangent theta".

Memorization Aid for the Sine, Cosine, and Tangent (there may be others):

SOHCAHTOA (Sine equals Opposite over Hypotenuse; Cosine equals Adjacent over Hypotenuse; and Tangent equals Opposite over Adjacent).

Cosecant Ratio:Pronounce

as "cosecant theta" (koseekent theta).csc

NOTE: The Cosecant Ratio is the reciprocal of the Sine Ratio!

Secant Ratio:Pronounce

as "secant theta" (seekent theta).sec

NOTE: The Secant Ratio is the reciprocal of the Cosine Ratio!

Cotangent Ratio:Pronounce

as "cotangent theta".cot

NOTE: The Cotangent Ratio is the reciprocal of the Tangent Ratio!

YOU MUST MEMORIZE THE SIX TRIGONOMETRIC IDENTITIES AS STATED ABOVE!

The Reciprocal and Quotient Identities- see #1 through 4in the "Examples" documentIn trigonometry, a great deal of time is spent studying relationships between trigonometric ratios. We call these relationships "identities."

Reciprocal Identities- Let's use these identities to define the cosecant, secant, and cotangent ratios.

Quotient Identities- Let's use these identities to define the tangent and cotangent ratios in terms of the sine and cosine ratios.

YOU MUST MEMORIZE THE RECIPROCAL AND QUOTIENT IDENTITIES!