Transcript 7.3 Sine & Cosine FCTNS

7.3 Sine & Cosine FCTNS
Objective
To use the definitions of sine and cosine to find values of these
functions and to solve simple trigonometric equations.
Suppose P( x, y) is a point on the circle x 2  y 2  r 2 and  is
an  in standard position with terminal OP.
y
x
sin  
& cos  
r
r
7.3 Sine & Cosine FCTNS
If the terminal ray passes through (  3,2), find
sin & cos .
Radius ?
2

3

2

r
   
2
2
9  4  r2
 r  13
2
2 13
sin  

13
13
3
3 13
cos  

13
13
7.3 Sine & Cosine FCTNS
5
If  is a 4th Quadrant  and sin   ,
13
find cos .
x?
x   5   13
2
2
2
x 2  25  169
 x 2  144
x 12
 x  12 4th Quadrant  x  12 cos   
r 13
7.3 Sine & Cosine FCTNS
Although the definitions of sine and cosine involve the
radius r of a circle, the values of sin and cos depend
only on  .
p. 269
Because all right 's that have the same given measure
for  are similar (corresponding 's  , corresponding
sides proportional) the value of a trigonometric ratio depends
only on the measure of  . It does not depend on the 's size.
y
y opp.
sin   

r hyp.
2 y
 y 2
2
2
y
2
sin   
r
2
2/2
2

1
2
1
1

2
2
y
x
7.3 Sine & Cosine FCTNS
(3, 4)
5

4
3
Find sin and cos .
SohCahToa
4 3
,
5 5
7.3 Sine & Cosine FCTNS
3
4

5
(3, 4)
4 3
Find sin and cos .
,
5 5
SohCahToa
7.3 Sine & Cosine FCTNS
Domain of sin & cos :  in
Range of sin & cos : y in -1,1
7.3 Sine & Cosine FCTNS
sin 90  ?
7.3 Sine & Cosine FCTNS
sin 450  ?
7.3 Sine & Cosine FCTNS
cos     ?
p. 270
7.3 Sine & Cosine FCTNS
Did you notice that sin 90  1 & sin 450 =1 ???
Solve sin   1 for  in degrees.
  90 ,   90  360 ,   90  2 360 ,   90  3 360 ,....
Degrees :
  90  n 360 , n in
Radians :


2
 n 2 or  

2
 2n , n in
So the sine and cosine functions repeat their values
every 360 or 2 radians.
 sin(  360 )  sin 
cos(  360 )  cos 
sin(  2 )  sin 
cos(  2 )  cos 
The sine and cosine functions are periodic. They have a
fundamental period of 360 or 2 radians. It's the periodic
nature of these functions that makes them useful in describing
many repetitive phenomena such as tides, sound waves, and
the orbital paths of satellites.
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