Triangle Definition of sin and cos

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.


(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


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 .


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


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


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


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=


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.


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


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=


xy sin21

=

xy sin=

xy sin41

=

x

y

-2

-1

0

1

2

π−π−2π 2π


xy sin=

π−π−2π 2πx

y

-2

-1

0

1

2xy sin−=


xy sin31−=

xy sin2−=

xy sin=

π−π−2π 2πx

y

-2

-1

0

1

2


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


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 .


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)


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


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


x

y

-5

-4

-3

-2

-1

0

1

2

3

4

5

y = sec x

25π

− 2

3π−

2π

− 2π

2

3π

25π


x

y

-5

-4

-3

-2

-1

0

1

2

3

4

5

y = cot x

−3π −2π −π π 2π 3π


x

y

-5

-4

-3

-2

-1

0

1

2

3

4

5

y = csc x

−3π −2π −π π 2π 3π

Triangle Definition of sin and cos

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