# Sec 2 Trigonometric Functions

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"Sec 2 Trigonometric Functions"

```					Sec 2 Trigonometric Expressions
Defn: If t is a real number that represents the
distance traveled along the unit circle and P(x, y)
is the terminal point on the unit circle, then each
trigonometric expression is defined as follows:

*the sine is defined to be sint = y
*the cosine is defined to be cost = x
*the tangent is defined to be tant = y/x
*the cosecant is defined to be csct = 1/y
*the secant is defined to be sect = 1/x
*the cotangent is defined to be cott = x/y

To evaluate a trigonometric expression:
1) Move the distance t on the unit circle
2) Locate the terminal point P associated with t
3) Use the definition and P(x,y) to find the value
4) The quadrant will determine the sign of the
value, unless the point is on an axis.
*REDUCE ALL RATIOS
Ex: Find the exact value of each expression
without using a calculator.
a) sin 
2

b) cos (   )
2
3
c) tan
2

d) sec 0
e) cot (  )
2
f) cos
3

g) tan (  )
3

h) sin 19
4

i) cos (  )
4
7
j) sec
6
11
k) cot
3

The value of trig expressions can also be found if
only the terminal point is given. The distance will
be represented as t. (REDUCE ALL RATIOS)

Ex: Find the sint, cost, and tant by using the given
terminal point determined by t.
a) P ( 12 , 5 )
13 13

5 61 6 61
b) P   (      ,     )
61   61

Ex: Find the quadrant in which the terminal point
determined by t lies.
a) sint < 0 and cost > 0
b) sint > 0 and tant > 0
c) sect < 0 and tant < 0
The calculator may be used to evaluate other
trigonometric expressions. If t is given in terms of
a real number then the mode must be set to the

Ex: Evaluate (set your calculator mode)
a) sin 2.2
b) cos 1.1
c) cot 28
d) csc .98

Fundamental Identities of Trig Expressions:

sin
1) tant = costt

2) cott =   cost
sin t

3) sin2t + cos2t = 1
4) tan2t + 1 = sec2t

5) 1 + cot2t = csc2t

6) csct = 1
sint

1
7) sect = cost

1
8) cott = tant

Ex: Use the fundamental identities to evaluate the
remaining trigonometric expressions.
3
a) cost = and t lands in Q4
5
b) cos t =   1
and   t  3
3                2

3
c) csc t = -3 and         t  2
2

4
d) cot t =        and sin t > 0
3
*There are two ways to show if a function is even
or odd:

* If f(-x) = -f(x) then f(x) is odd.
* If f(x) is swrt the origin then f(x) is odd.

* If f(-x) = f(x) then f(x) is even.
* If f(x) is swrt the y-axis then f(x) is even.

The algebra test:
1) replace x with –x
2) simplify the expression
3) analyze the result.

Summary of trig expressions:
sin (-t) = -sin t  csc (-t) = -csc t
cos (-t) = cos t   sec (-t) = sec t
tan (-t) = -tan t  cot (-t) = -cot t
Ex: Determine if the function is even, odd or
neither.
a) f(x) = x2 cosx

b) f(x) = x2 tanx

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