1.3 definition 1 of trigonometric functions jmerrill, 2009

1.3 Definition 1 of Trigonometric

Functions

JMerrill, 2009

Trigonometry The word trigonometry comes from

two Greek words, trigon and metron, meaning “triangle measurement”. We will “measure” triangles by concentrating on their angles.

Definition 1 ONLY works for right triangles

Trigonometric Functions (Ratios) There are six trigonometric functions:

Sine abbreviated sin--sinθ Cosine abbreviated cos--cosθ Tangent abbreviated tan--tanθ Cosecant abbreviated csc--cscθ Secant abbreviated sec--secθ Cotangent abbreviated cot--cotθ

Recall from 1.2 We discussed the ratios of the sides

of similar triangles The three main trigonometric

functions should be learned in terms of the ratios of the sides of a triangle.

Right Triangle Trig

SOH-CAH-TOA Sin θ = Cos θ = Tan θ =

These are the ratios of 2 sides with respect to an angle.

In order to find the other trig functions, we must look at some identities

ppositeypoteO

H nuse

djacentypoteA

H nuse

OA

ppositedjacent

θ

oppositehypotenuse

adjacent

Fundamental Trigonometric IdentitiesReciprocal Identities

1csc

sin

1

seccos

1cot

tan

Also true:

1sin

csc

1

cossec

1tan

cot

Example Find the following—exact answers

only D

4 5 Sin D = Sin G =

Cos D = Cos G =

O 3 G Tan D = Tan G =

35

4534

453543

Board Example

Cofunctions Notice the co in cosine, cosecant, and

cotangent. These are cofunctions and they are based on the relationship of complementary angles.

The Cofunction Theorem states that if α+β = 90o, then: sin β = cos α

sec β = csc αtan β = cot α

Cofunction Examples Sin 30o =

Csc 40o =

Tan x =

Cos 60o

Sec 50o

Cot (90o-x)

Fundamental Trigonometric Identities

Cofunction Identities

sin cos 90o cos sin 90o

tan cot 90o cot tan 90o

sec csc 90o csc sec 90o