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
Cosine
Tangent
Cosecant
Secant
Cotangent





abbreviated sin--sinθ
abbreviated cos--cosθ
abbreviated tan--tanθ
abbreviated csc--cscθ
abbreviated sec--secθ
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 =
Opposite
Hypotenuse

Cos θ =
Adjacent
Hypotenuse

Tan θ =
Opposite
Adjacent


hypotenuse
opposite
θ
adjacent
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
Fundamental Trigonometric
Identities
Reciprocal Identities
sin  
1
csc
1
cos 
sec
1
tan  
cot 
Also true:
1
csc 
sin 
1
sec 
cos
1
cot  
tan 
Example

Find the following—exact answers only
D
4
5
Sin D =
Cos D =
O
3
G
Board Example
Tan D =
3
5
4
5
3
4
Sin G =
Cos G =
Tan G =
4
5
3
5
4
3
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 α
tan β = cot α
Cofunction Examples

Sin 30o =
Cos 60o

Csc 40o =
Sec 50o

Tan x =
Cot (90o-x)
Fundamental Trigonometric
Identities
Cofunction Identities
sin   cos  90o   
cos  sin  90o   
tan   cot  90   
cot   tan  90o   
sec  csc  90o   
csc  sec  90o   
o