Math 1016: Elementary Calculus with Trigonometry I
Trigonometric Functions and Graphs
I.
Trig Functions
A. Trig Functions and the Unit Circle
Choose
P(x, y) a point on the unit circle where the
terminal side of θ intersects with the circle.
Then cosθ = x and sin θ = y .
We see that the Pythagorean Identity follows directly
from these definitions:
x2 +
y2 = 1
cos2 θ + sin 2 θ = 1
B. 6 Trig Functions
There are six trigonometric functions. We have considered the sine and cosine
functions. We can define the four remaining in terms of these functions.
tan θ =
The tangent function:
The cosecant function:
sin θ
cosθ
csc θ =
1
sin θ
The cotangent function:
The secant function:
cot θ =
sec θ =
cos θ
sin θ
1
cos θ
We know all of the above functions will have points of discontinuity where the
denominator is zero. The graphs of these functions all have vertical asymptotes at
these points.
C. Common angles
degrees
00
radians
0
sin
cos
θ
0
300
π
6
1
2
θ
1
3
2
The common angles
450
π
4
600
π
3
1
2
=
2
2
1
2
=
2
2
3
2
1
2
900
π
2
1800
3600
π
2700
3π
2
1
0
-1
0
0
-1
0
1
< 90° listed above will become reference angles.
2π
D. Signs of Trig Functions
Using the definition of the sine and cosine functions on the unit circle we can find the
signs of the trigonometric functions in each quadrant.
cosθ = x
and
sin θ = y
E. Examples
Evaluate the following without using a calculator
a.
⎛ 3π ⎞
sin⎜− ⎟
⎝ 4 ⎠
b.
⎛ 29π ⎞
cos⎜
⎟
⎝ 6 ⎠
c.
tan(420°)
d.
⎛ 9π ⎞
sec⎜− ⎟
⎝ 4 ⎠
e.
csc(510°)
f.
⎛19π ⎞
cot⎜
⎟
⎝ 3 ⎠
x in
x in
sin(x) cos(x)
degrees radians
00
0
π
6
0
1
2
60
0
π
3
3
2
90
0
1
0
135
150
0
5π
6
2
2
1
2
π
7π
6
0
1
−
2
1800
2100
225
0
5π
4
−
2
2
240
0
4π
3
−
3
2
270
0
300
0
−
=
2
2
−
3
2
-
-1
−
-1
3
2
2
2
1
−
2
−
1
3
=−
3
3
1
2
− 3
315
0
7π
4
−
2
2
2
2
-1
330
0
11π
6
1
2
3600
2π
3
2
1
0
−
-
−
1
3
=
3
3
1
-
0
-2
3
2
−
2
1
3
=
3
3
−
0
2
2 3
=
3
3
-
3
1
3
=−
3
3
0
-
1
0
1
1
cos( x )
sin( x )
2
= 2
2
2
−
1
cos( x )
2
2 3
=
3
3
2
= 2
2
2
2 3
=
3
3
2
= 2
2
− 3
3π
2
5π
3
-1
-
3
1
2
−
−
=
1
sin( x )
1
45
2
2
1
2
3π
4
=
sin( x )
cos( x )
2
2
2
0
=
1
3
=
3
3
π
4
120
cot(x)
3
2
0
3
2
sec(x)
0
30
0
csc(x)
1
0
π
2
2π
3
tan(x)
-2
−
−
3
−
1
3
=−
3
3
2
=− 2
2
-1
2
2 3
=−
3
3
− 3
-1
-
2
=− 2
2
2
2 3
=−
3
3
2
−
=− 2
2
1
2
2 3
=−
3
3
-2
1
3
=
3
3
-
0
-1
2
2 3
=−
3
3
2
−
=− 2
2
−
-2
-
−
2
2
= 2
2
2
2 3
=
3
3
1
3
−
1
3
=−
3
3
-1
− 3
-
Special Angles on the Unit Circle
y
(
(
(
− 3
2
, 12
− 2
2
,
2
2
−1
2
,
3
2
( 0,1)
)
(
1
3
2
1
)
)
3
2
)
2
2
,
2
2
)
600
1
2
1350
,
(
2
1200
1
2
(
3
2
, 12
)
0
45
1500
900
1800
( −1, 0)
1
3
-1 2
2
300
00
3600
1
2
1
2
1
(1,0 )
3
2
2
1
2700
2100
(
3
2
, −12
)
-
0
225
0
240
(
− 2
2
, −22
)
(
II.
−1 − 3
2
2
,
)
-
3300
1
2
(
3150
− 3
2
, −12
0
300
1
2
3
2
-1
( 0,−1)
(
(
1
2
, −2 3
2
2
, −22
)
)
Graphs of Trig Functions
A.
y =Asin(bx) + C or y=Acos(bx) + C
|A| is the amplitude: how high/low the graph goes from the center of the graph.
normal period 2π
2.
is the period: the length of the interval before the graph
=
b
b
repeats its shape. For y =tan(x) and y =cot(x) the normal period is π.
3. C is a vertical shift: upward/downward shift.
1.
)
x
B. Examples
1.
x
−2π
−
y =sin(x)
11π
6
−
3π
2
−
−π
7π
6
−
5π
6
−
π
2
−
π
6
π
6
0
5π
6
π
2
π
7π
6
3π
2
11π
6
y=sinx
amplitude:
period:
vertical shift:
y
1
-2 p
-
3p
2
-
-p
p
2
p
2
x
3p
2
p
2p
-1
2.
y=sin(2x)
−
−2π
x
3π
2
−
5π
4
−
−π
3π
π
−
4
2
−
π
4
0
3π
4
π
2
π
4
π
5π
4
3π
2
2π
y=sinx
amplitude:
period:
vertical shift:
y
1
-2 p
-
7p
4
-
3p
2
-
5p
4
-p
-
3p
4
-
p
2
-
p
4
p
4
-1
p
2
3p
4
p
5p
4
3p
2
7p
4
x
2p
2π
3.
y =2sin(x)
−
−2π
x
3π
2
−π
−
π
2
0
π
2
3π
2
π
2π
y=sinx
amplitude:
period:
vertical shift:
y
2
1
-
-2 p
3p
2
-
-p
p
2
p
2
3p
2
p
x
2p
-1
-2
4.
y =2sin(x)-1
−
−2π
x
3π
2
−π
−
π
2
π
0 2
3π
2
π
2π
y=sinx
amplitude:
period:
vertical shift:
y
2
1
-2 p
-
3p
2
-p
-
p
2
p
2
-1
-2
-3
p
3p
2
x
2p
C. Basic graphs of the 6 Trig Functions on
1.
y =sin(x)
[−2π , 2π ]
y =cos(x)
2.
y á cosHxL
y á sinHxL
y
y
1
1
-2 p -
3p
2
-p
-
p
2
p
2
3p
2
p
x
-2 p -
2p
3p
2
-p
-
p
2
y =tan(x)
y
1
7p 3p 5p
3p p p
- -p - - - -1
4 2 4
4
2 4
p
4
p
2
3p
4
p
5p 3p 7p
2p
4 2 4
y =sec(x)
1
x
-2 p
6.
y á secHxL
-
3p
2
-p
-
p
2
-1
3p
2
-p
-
p
2
-1
x
2p
y á cotHxL
y
1
1
-
3p
2
y =cot(x)
y
-2 p
p
x
2p
y á cscHxL
y
5.
p
2
3p
2
y =csc(x)
4.
y á tanHxL
-2 p-
p
-1
-1
3.
p
2
p
2
p
3p
2
x
2p
7p 3p 5p
3 p p p-1
-2 p- - -p- - 4 2 4
4 2 4
p
4
p 3p
2 4
p
5p 3p 7p
2p
4 2 4
x