1 trigonometric functions of any angle & polar coordinates sections 8.1, 8.2, 8.3, 21.10
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2
Definitions of Trigonometric Functions of Any Angle
Let be an angle in standard position with (x, y) a point on the terminal side of and
Definitions of Trig Functions of Any Angle
2 2r x y
sin csc
cos sec
tan cot
y r
r y
x r
r xy x
x y
y
x
(x, y)
r
3
Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.
The Signs of the Trig Functions
5
Where each trig function is POSITIVE:
A
CT
S
“All Students Take Calculus”
Translation:
A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4
*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tan is positive, but sine and cosine are negative; ...
**The reciprocal functions have the same sign. So cosecant is positive wherever sine is positive, secant is positive wherever cosine is positive, …
6
Determine if the following functions are positive or negative:
Example
sin 210°
cos 320°
cot (-135°)
csc 500°
tan 315°
7
sin csc
cos sec
tan co
0
t
y r
r y
x r
r xy x
x y
r
Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the
axes , we will use the circle.
(..., 180 , 90 , 0 , 90 , 180 , 270 , 360 ,...)
(0, r) 90
(r, 0)(-r, 0)
(0, -r)
0
270
180
8
Find the value of the six trig functions for
Example
90
(r, 0)
(0, r)
(-r, 0)
(0, -r)
0
270
90
180
sin 90
cos 90
tan 90
csc 90
sec 90
cot 90
y
rx
ry
xr
y
r
xx
y
9
The values of the trig functions for non-acute angles (Quads II, III, IV) can be found using the values of the corresponding reference angles.
Reference Angles
Definition of Reference Angle
Let be an angle in standard position. Its reference angle is the acute angle formed by the terminal side of and the horizontal axis.
'
10
Example
Find the reference angle for 225
Solution y
x
'
By sketching in standard position, we see that it is a 3rd quadrant angle. To find , you would subtract 180° from 225 °.
'
225 180
45
'
'
11
So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we just need to find the trig function of the reference angle and then determine whether it is positive or negative, depending upon the quadrant in which the angle lies.
For example,
1sin 225 (sin 45 )
2
45° is the ref angleIn Quad 3, sin is negative
13
Now, of course you can simply use the calculator to find the value of the trig function of any angle and it will correctly return the answer with the correct sign.
Remember:
Make sure the Mode setting is set to the correct form of the angle: Radian or Degree
To find the trig functions of the reciprocal functions (csc, sec, and cot), use the button.
14
Example
Evaluate the trig function to four decimal places.
Set Mode to Degree
Enter: 324
cot 324
: cot 324 1.3764ANS
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HOWEVER, it is very important to know how to use the reference angle when we are using the inverse trig functions on the calculator to find the angle because the calculator may not directly give you the angle you want.
r-5
y
x
(-12, -5)
-12
Example: Find the value of to the nearest 0.01°
17
In general, for in radians,
A second way to measure angles is in radians.
Radian Measure (Sect 8.3)
s
r
Definition of Radian:
One radian is the measure of a central angle that intercepts arc s equal in length to the radius r of the circle.
18
Radian Measure
2 radians corresponds to 360
radians corresponds to 180
radians corresponds to 902
2 6.28
3.14
1.572
20
Conversions Between Degrees and Radians
1. To convert degrees to radians, multiply degrees by
2. To convert radians to degrees, multiply radians by
180
180
Example
Convert from Degrees to Radians: 210º
210
21
Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
b) Convert from radians to degrees: 3.8
3
4
3
4
3.8
22
Conversions Between Degrees and Radians
c) Convert from degrees to radians (exact):
d) Convert from radians to degrees:13
6
13
6
675
675
23
Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places):
f) Convert from radians to degrees (to nearest tenth): 1 rad
5252
1
25
Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an ordered pair .
• If , then r is the distance of the point from the pole.
is an angle (in degrees or radians) formed by the polar axis and a ray from the pole through the point.
,r
0r
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Polar Coordinates
If , then the point is located units on the ray that extends in the opposite direction of the terminal side of .0r r
33
To find the rectangular coordinates for a point given its polar coordinates, we can use the trig functions.
4,3
Example
35
Likewise, we can find the polar coordinates if we are given the rectangular coordinates using the trig functions.
Example:
Find the polar coordinates for the point for which the rectangular coordinates are given: (5, 4) Express answer to three sig digits.
(5, 4)
36
Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates (r, ) of P are given by
2 2 1tanref
yr x y
x
P
You need to consider the quadrant in
which P lies in order to find the value
of .
39
The TI-84 calculator has handy conversion features built-in. Check out the ANGLE menu.
5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of given rectangular coordinates (x, y)
7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )
Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
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