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Transcript
Chapter 3
Trigonometric Functions of Angles
Section 3.3
Trigonometric Functions of Angles
y-axis
Angles in Standard Position
Recall an angle in standard position is an angle
that has its initial side on the positive x-axis. We
can use any point on the angles terminal side to
find the values of the trigonometric ratios. If the
coordinates of the point P are (x,y) and the
distance the point P is from the origin is r we get
the following values for the trigonometric ratios.
y
sin   
r
cos  
x

r
y
tan  
x
x
cot  
y
y
x2  y2
x
x2  y 2
P:(x,y)
r  x2  y2
y

x
r
sec   
x
x2  y2
x
r
csc   
y
x2  y 2
y
In the example to the right with the coordinates
of P at the point (1,3)
3
10
1
cos  
10
sin  
3
tan  
1
cot  
1
3
10
sec  
1
csc  
10
3
x-axis
P:(1,3)
3
2 r
1 
1 2
r  12  32
r  10
It is important to realize that it does not matter what point you
select on the terminal side of the angle the trigonometric
ratios will be the same because the triangles are similar. The
triangle with its vertex at P1 is similar to the triangle with its
vertex at P2 and the length of the sides are proportional
(equal ratios).
P1
P2

Signs of Trigonometric Functions
The trigonometric ratios now are defined no matter where
the terminal side of the angle is. It can be in any if the four
quadrants. Since the values for the xy-coordinates are
different signs (±) depending on the quadrant the
trigonometric ratios will be also. The value for r is always
positive. The chart below shows the signs of the
trigonometric ratios.
x neg (-)
y pos (+)
x pos (+)
y pos (+)
x neg (-)
y neg (-)
x pos (+)
y neg (-)
Quadrant
sin
cos
tan
cot
sec
csc
I
+
+
+
+
+
+
II
+
-
-
-
-
+
III
-
-
+
+
-
-
IV
-
+
-
-
+
-
Find the values of the six trigonometric functions if the point (-3,2)
is on the terminal side of the angle. We find the value for r
(distance from the origin) first. r  (3) 2  2 2  9  4  13
2
r

-3
2
13
3
cos  
13
sin  
tan  
2
3
sec  
13
3
cot  
3
2
csc  
13
2
Reference Angles
The reference angle for an angle is the angle made when you drop a line
straight down to the x-axis. it is the angle made by the x-axis regardless of what
side of it you are on.
225
120
330
60
60
45
30
-300
Reference angles are useful to help you find the
values for trigonometric functions for many angles
of the circle. For example if we want to find the
trigonometric ratios for 150. We know the
reference angle is 30 and we for a 30-60-90
triangle. The sides are in the ratios we mentioned
before.
1

sin 150 
2
 3
cos150 
2
1
1
2
150
30
 3
2
1
tan 150 
3
sec 150 
cot 150   3
csc 150  2
2


 3
Identities
csc  
1
sin 
sec  
sin 2   cos 2   1
1
cos 
tan  
1
cot 
tan  
sin 
cos 
cot  
cos 
sin 
tan 2   1  sec 2  1  cot 2   csc 2 