Post on 10-May-2023
MATH 1380 Lecture 4 1 of 21 Ronald Brent © 2018 All rights reserved.
(1,0)
(cos θ , sin θ )
t
y
x
Consider the unit circle, centered at the origin, with an angle of t radians, as shown below.
(Notice that the angle is measured from the positive x-axis, counterclockwise.) The dotted line
defining the terminal side (end) of the angle t intersects the circle at a point. As the angle t changes,
so do the coordinates of that point, so each of the coordinates is a function of the angle t. These two
functions are very important, and so they have their own names.
MATH 1380 Lecture 4 2 of 21 Ronald Brent © 2018 All rights reserved.
(1,0)
(x, y) = (cos t , sin t )
t
y
x
Definition: In the figure below, the first coordinate is called cos t (short for cosine of t ). The second
coordinate is called sin t ( short for sine of t).
t
MATH 1380 Lecture 4 3 of 21 Ronald Brent © 2018 All rights reserved.
In terms of x and y.
yt=sin xt=cos xyt=tan
yt 1sec = x
t 1csc = yxt=cot
Remarks:
a) Since this point is on the unit circle, its coordinates must satisfy the equation of that circle:
122 =+ yx , that is 1)(sin)(cos 22 =+ tt .
b) To avoid the constant use of brackets, we write tncos to mean nt)(cos ; similarly, we write
tnsin to mean nt)(sin . Thus 1sincos 22 =+ tt .
c) Since the cycle repeats every time we go around the circle, the sine and cosine functions
are periodic with period π2 .
MATH 1380 Lecture 4 4 of 21 Ronald Brent © 2018 All rights reserved.
Table of Trig. Values:
Since 6
30 π= radians,
445 π
= radians, and 3
60 π= radians, we can use the previous triangle trig.
results and the following picture to fill in the table on the next page.
(1,0)x
y
(0,1)
(-1,0)
(0,-1)
)sin,cos( tt
)0sin,0cos(
)sin,cos( 23
23 ππ
)sin,cos( ππ
)sin,cos( 22ππ
MATH 1380 Lecture 4 5 of 21 Ronald Brent © 2018 All rights reserved.
Angle t
ty sin=
tx cos=
0 0 1
6π
21
23
4π
22
22
3π
23
21
2π
1 0
π 0 −1
23π
−1 0
π2 0 1
MATH 1380 Lecture 4 6 of 21 Ronald Brent © 2018 All rights reserved.
π 2πx
−2
−1
0
1
2
π 2πx
−2
−1
0
1
2
(1,0)
(cos t , sin t )
t
y
x
Graphs of the sine and cosine functions
The second coordinate, ty sin= , goes from
0 to 1, and back down to −1, then back up to
1.
Meanwhile, the first coordinate, tx cos= ,
goes from 1 down to −1, and back to 1.
ty sin=
tx cos=
t
t
MATH 1380 Lecture 4 7 of 21 Ronald Brent © 2018 All rights reserved.
Graphs of Sine and Cosine (Sinusoidal) Functions Going back to y as a function of x
Notice how these graphs oscillate between 1 and −1. Also, the length, or period, of one full
cycle is π2 .
2π 4π− 2π
−2
−1
1
2
x
2π 4π− 2π
−2
−1
1
2
x
xy sin=
xy cos=
MATH 1380 Lecture 4 8 of 21 Ronald Brent © 2018 All rights reserved.
Formal Definitions
Any function of the form ktbay += sin or ktbay += cos is called sinusoidal.
The Amplitude of a sinusoidal graph is equal to one-half the distance from the top to the bottom
of the waves, or the number |a|.
The Period of a sinusoidal function is the distance for the graph to go through one full cycle.
It is alwaysb
P π2= .
The Angular (Circular) Frequency of a sinusoidal function, |b|, is the number of complete cycles in a horizontal distance of π2 . The Linear Frequency , f is the reciprocal of the period, so
that π2bf = . If t represents time, then f has units of cycles per second.
The horizontal line y = k, is called the center line about which the function oscillates.
MATH 1380 Lecture 4 9 of 21 Ronald Brent © 2018 All rights reserved.
Example: The graph below is 32sin2 += ty . Its amplitude is 2|2||| ==a , (NOT the
bigger number 5.) The circular frequency is 2, the period is π , and the frequency is π1
The center line is y = 3.
x
y
-2
-1
0
1
2
3
4
5
−π 2ππ−2πt
MATH 1380 Lecture 4 10 of 21 Ronald Brent © 2018 All rights reserved.
x
y
-5
-4
-3
-2
-1
0
1
2
3
4
5
π−π−2π 2π
Changes in Amplitude:
xy sin2=
xy sin4=
xy sin=
MATH 1380 Lecture 4 11 of 21 Ronald Brent © 2018 All rights reserved.
xy sin21
=
xy sin=
xy sin41
=
x
y
-2
-1
0
1
2
π−π−2π 2π
MATH 1380 Lecture 4 12 of 21 Ronald Brent © 2018 All rights reserved.
xy sin=
π−π−2π 2πx
y
-2
-1
0
1
2xy sin−=
MATH 1380 Lecture 4 13 of 21 Ronald Brent © 2018 All rights reserved.
xy sin31−=
xy sin2−=
xy sin=
π−π−2π 2πx
y
-2
-1
0
1
2
MATH 1380 Lecture 4 14 of 21 Ronald Brent © 2018 All rights reserved.
x
y
-5
-4
-3
-2
-1
0
1
2
3
4
5
x
y
-5
-4
-3
-2
-1
0
1
2
3
4
5
x
y
-5
-4
-3
-2
-1
0
1
2
3
4
5
x
y
-5
-4
-3
-2
-1
0
1
2
3
4
5
Of course all this vertical amplitude scaling works for the cosine graph also.
xy cos2=
xy cos21−=
xy cos21=
xy cos4−=
−2π −π π 2π
MATH 1380 Lecture 4 15 of 21 Ronald Brent © 2018 All rights reserved.
Changes in Frequency: Going from y = sin x or y = cos x, to )(sin bxy = and )(cos xby = involves horizontal scaling.
This affects how many cycles appear over a given interval. As a rule:
(a) If b is a positive integer, then the graph of )(sin xby = ( )(cos xby = ) has b complete
oscillations, or cycles, in the interval ]2,0[ π . For b positive, if b > 1 this means more
oscillations than y = sin x (y = cos x) and for 0 < b < 1, one has less cycles than y = sin x (y
= cos x).
(b) If b < 0, the graph is reflected about the y-axis, and then compressed or stretched depending
upon the value of |b|.
Note: )(sin xby = and xby sin= are NOT the same.
Test it with b = 2 and2π
=x .
MATH 1380 Lecture 4 16 of 21 Ronald Brent © 2018 All rights reserved.
π−π−2π 2πx
y
-2
-1
0
1
2
π−π−2π 2πx
y
-2
-1
0
1
2
π−π−2π 2πx
y
-2
-1
0
1
2
Again, in all of these graphs xy sin= is shown as a solid line.
Period = π Period = 3
2π
Period = 4π
xy 2sin= xy 3sin=
y = sin ( x/2)
MATH 1380 Lecture 4 17 of 21 Ronald Brent © 2018 All rights reserved.
Graphs of Other Trigonometry Functions
Tangent: tansincos
xxx
= Undefined if 0cos =x , or π
+
=2
12 nx
Cotangent: cotcossin
xxx
= Undefined if 0sin =x , or πnx =
Secant: seccos
xx
=1
Like Tangent, undefined for π
+
=2
12 nx
Cosecant: cscsin
xx
=1
Like Cotangent, undefined for πnx =
In all the above expressions, ,2,1,0 ±±=n
MATH 1380 Lecture 4 18 of 21 Ronald Brent © 2018 All rights reserved.
x
y
-5
-4
-3
-2
-1
0
1
2
3
4
5
Graphs of Other Trigonometry Functions
y = tan x
25π
− 2
3π−
2π
− 2π
2
3π
25π
MATH 1380 Lecture 4 19 of 21 Ronald Brent © 2018 All rights reserved.
x
y
-5
-4
-3
-2
-1
0
1
2
3
4
5
y = sec x
25π
− 2
3π−
2π
− 2π
2
3π
25π
MATH 1380 Lecture 4 20 of 21 Ronald Brent © 2018 All rights reserved.
x
y
-5
-4
-3
-2
-1
0
1
2
3
4
5
y = cot x
−3π −2π −π π 2π 3π