Download Triangle Definition of sin θ and cos θ ADJ (side adjacent to the angle

Triangle Definition of sin  and cos 
Consider the triangle ABC below. Let  A be called  .
B
HYP
(hypotenuse)
A
OPP
(side opposite to
the angle )

C
ADJ
(side adjacent to
the angle )
Then
sin  
OPP
BC

HYP
AB
cos  
ADJ
AC

HYP
AB .
(SOH CAH TOA)
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Lecture 4A
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Special Angles (30°, 45°, 60°)

1

Find sin 45 and cos 45 .
2
2
2
Pythagorean Theorem: s  s  1 , or 2 s  1
Hence
Choosing s > 0, we have s 
1
2

s
45
1
1
.
s 2  and s  
2
2
sin 45   cos 45  
45
s
1
, and so
2
2
2
45
1
1
2
45
1
2
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Lecture 4A
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Example:




Find sin 30 , cos 30 , sin 60 , and cos 60 .



Begin with a 30  60  90 right triangle with HYP = 1.
60
1
t
30
s
30
1
60
Notice that when flipping the triangle down and consider the larger one, the result is an equilateral
1
triangle, so the vertical side is also 1. This means 2t = 1, or t  . Now since
2
s2  t 2  1 , we have
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1
s2   1 ,
4
Lecture 4A
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which means
s2 
3
3
 s
4
2
Again, it is clear that s > 0, which implies the triangle looks like:
1
60
1
2
30
3
2
So
sin 30  
1
2
cos30 
3
,
2
while
cos 60  
1
2
sin 60 
3
.
2
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Ronald Brent © 2016 All rights reserved.
Lecture 4A
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Angles in Radian Measure
How big is a radian? Here’s how big: it’s the angle corresponding to an arc length of 1 in a unit
circle. Look at the diagram below. A “unit circle” indicates that the radius = 1 unit, and we’ll
always put the center at (0,0) for convenience. The angle as drawn is 1 radian, because the arc
length “subtended” (cut off) by the angle has length = 1 unit.
Length of the arc = 1 unit

radian
(0,0)
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Lecture 4A
1 unit
(1,0)
5 of 25
Relationship Between Degrees and Radians
360   2 radians
360
1 rad = 2 
1 
Examples:

 57
2

rad =
rad  0.017 rad .
360
180

a) Convert 30 to radians.
1 

180
Note:
radians 
45 
30   30 


180

6
.
 , 60   and 90   .
4
3
2
5
b) Convert
radians to degrees.
6
  180
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
Lecture 4A
5
5
    180   150  .
6
6
6 of 25
You should commit to memory the following conversion chart since these angles will come up
again and again.
Degrees
360
270
180
Radians
2
3
2

90
60
45
30




2
3
4
6
Consider the unit circle, centered at the origin, with an angle of  radians, as shown below.
y
(cos  , sin  )

(1,0)
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Lecture 4A
x
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(Notice that the angle is measured from the positive x-axis, counterclockwise.) The dotted line
defining the terminal side (end) of the angle  intersects the circle at a point. As the angle 
changes, so do the coordinates of that point, so each of the coordinates is a function of the angle .
These two functions are very important, and so they have their own names.
Definition: In the figure below, the first coordinate is called cos  (short for cosine of  ). The
second coordinate is called sin  ( short for sine of  ).
y
(cos  , sin  )

(1,0)
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Lecture 4A
x
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Remarks:
a) Since this point is on the unit circle, its coordinates must satisfy the equation of that
circle: x 2  y 2  1 , that is (cos  ) 2  (sin  ) 2  1 .
n
n
b) To avoid the constant use of brackets, we write cos  to mean (cos  ) ; similarly, we
n
n
write sin  to mean (sin  ) . Thus cos2   sin 2   1.
c) Since the cycle repeats every time we go around the circle, the sine and cosine functions
are periodic with period 2  .
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Lecture 4A
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Table of Trig. Values:
Since 30 

6
radians, 45 

4
radians, and 60 

3
radians, we can use the previous triangle trig.
results and the following picture to fill in the table on the next page.
y
( cos  , sin  )
(0,1)
(-1,0)
(1,0)
x
(0,-1)
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Lecture 4A
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Angle 
y  sin 
x  cos 
0
0
1

1
2
3
2
2
2
2
2
3
3
2
1
2

1
0

0
1
3
2
1
0
2
0
1
6

4

2
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Lecture 4A
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Trig Values at Other Angles:
Definition:
An Obtuse angle is any angle greater than
Example: Find sin
Since
2
2
and cos
.
3
3
2
 120 , the triangle in the
3
figure below is the
30  60  90


y
2
 120 
3
3
2
60
1
2
2
1
  .
3
2
Ronald Brent © 2016 All rights reserved.
radians.
1

2
3
Hence, sin

3
2
MATH 1310
2
 1 3
 ,

 2 2 
triangle shown previously.
and cos

Lecture 4A

x
(1,0)
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Definition:
A negative angle is an angle measured in the clockwise direction.
 
 
Example: Find sin   and cos   .
 3
 3
Since 

3
y
 60 , the triangle in the figure



shown is again the 30  60  90 triangle
1
2
shown above.
60
1
3
 
  1
Hence, sin    
and cos    .
2
 3
 3 2
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Lecture 4A
x

3
2
(1,0)
13 of 25
Graphs of the sine and cosine functions
y

(cos  , sin  )
y  sin 



(1,0)
x


x



The second coordinate, y  sin  , goes

from 0 to 1, and back down to 1, then back

up to 1.
y  cos


x

Meanwhile, the first coordinate, x  cos  ,

goes from 1 down to 1, and back to 1.
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Lecture 4A
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Graphs of Sine and Cosine (Sinusoidal) Functions
y  sin x


x
 




y  cos x


x





Notice how these graphs oscillate between 1 and 1. Also, the length, or period, of one full
cycle is 2  .
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Lecture 4A
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Formal Definitions
Any function of the form
y  k sin( a x   )  C
or
y  k cos( a x   )  C
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 |k|.
The Period of a sinusoidal function is the distance for the graph to go through one full cycle.
2
It is always P 
.
a
The Angular (Circular) Frequency of a sinusoidal function, |a|, is the number of complete
cycles in a horizontal distance of 2  . The Linear Frequency , f is the reciprocal of the
a
period, so that f 
. If x represents time, then f has units of cycles per second.
2
The Phase  of a sinusoidal function is what point in its cycle it “starts” at, when x = 0. It
represents horizontal shifts in the sinusoidal function.
The horizontal line y = C, is called the center line about which the function oscillates.
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Lecture 4A
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Example:
The graph below is f ( x)  2 sin 2 x  3 . Its amplitude is | k |  | 2 |  2 , (NOT the
bigger number 5.) The circular frequency is 2, the period is  , and the frequency is
1
. The phase is 0, and the center line is y = 3.

y
5
4
3
2
1


0


x
-1
-2
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Ronald Brent © 2016 All rights reserved.
Lecture 4A
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Changes in Amplitude:
y
y  4 sin x
5
y  2 sin x
4
y  sin x
3
2
1
 

0


x
-1
-2
-3
-4
-5
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Lecture 4A
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y
1
y  sin x
4
2
1
y  sin x
2
y  sin x
1
 

0


x
-1
-2
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Lecture 4A
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y
y  sin x
2
y   sin x
1
 

0


x
-1
-2
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Lecture 4A
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y
y  sin x
2
y   13 sin x
1
 

0


x
-1
y  2 sin x
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-2
Lecture 4A
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Of course all this vertical amplitude scaling works for the cosine graph also.
y
y  2 cos x
y
5
y  12 cos x
4
5
4
3
3
2
2
1
1
0

x
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
y
y  4 cos x
y
y   12 cos x
5
4
MATH 1310
5
4
3
3
2
2
1
1
0
x
0
-1
-1
-2
-2
-3
-3
-4
-4
-5
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x
0
Lecture 4A
-5
x
22 of 25
Changes in Frequency:
Going from y = sin x or y = cos x, to y  sin ( a x ) and y  cos ( a x ) involves horizontal scaling.
This affects how many cycles appear over a given interval. As a rule:
(a)
If a is a positive integer, then the graph of y  sin ( a x ) ( y  cos ( a x ) ) has a complete
oscillations, or cycles, in the interval [0, 2  ] . For a positive, if a > 1 this means more
oscillations than y = sin x (y = cos x) and for 0 < a < 1, one has less cycles than y = sin x
(y = cos x).
(b)
If a < 0, the graph is reflected about the y-axis, and then compressed or stretched
depending upon the value of |a|.
y  sin ( a x ) and y  a sin x are NOT the same.
Note:
Test it with a = 2 and x 
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
2
.
Lecture 4A
23 of 25
Examples: Again, in all of these graphs y  sin x is shown as a solid line.
y  sin 2 x
y  sin 3 x
y
2
Period =
3
2
Period = 
y
2
1
1


0
x


 
0

-1


-1
-2
-2
y
y = sin ( x/2)
2
Period = 4 

1

0


x
-1
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-2
Lecture 4A
24 of 25
x
Phase Shifts:
Phase shifts involve horizontal translations, of shifts in the x-direction.
Examples:

y  sin x  2


y  cos x  2
y
 

y
5
5
4
4
3
3
2
2
1
1
0


x
 

0
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
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Ronald Brent © 2016 All rights reserved.

Lecture 4A


x
25 of 25