Transcript Document

§1.6 Trigonometric Review
The student will learn about:
angles in degree and
radian measure,
trigonometric
functions, graphs of
sine and cosine
functions, and the
four other
trigonometric
functions.
1
§1.6 Trigonometric Review
What follows is basic information about the
trigonometric essentials you will need for
calculus. It is not complete and assumes you
have a full knowledge of trigonometric
functions.
2
Degrees and Radians
Angles are measured in degrees where there are 360º
in a circle.
Angles are also measured in radians where there are
2π radians in a circle.
Degree-Radian Conversion Formula
deg
rad


180
 rad
Example 1. Convert 90º to radians.


90
90

or  rad   rad
 rad


2
180
 rad
180

rad

3
Degrees and Radians
Degree-Radian Conversion Formula
deg
rad


180
 rad
Example 2. Convert π/3 radians to degrees.

 3rad

  3 rad

or   180
= 60 º

180
 rad
rad
Some Important Angles
Radian
0
π/6
π/4
π/3
π/2
π
2π
Degree
0º
30º
45º
60º
90º
180º
360º
4
Trigonometric Functions
Consider a unit circle with center at
the origin. Let point P be on the
circle and form an angle of θ (in
radians) with the positive x axis.
The cosine θ is the abscissa
of point p, i.e. x = cosine θ.
(0, 1)
P (x, y)
θ
(1, 0)
The sine θ is the ordinate of
point p, i.e. y = sine θ.
To find the value of either the sine or cosine functions
use the sin and cos keys of your calculator. Make sure
you are in the correct mode, [either degrees or radians]
pertaining to the problem.
5
Trigonometric Functions
Consider a unit circle with center at
the origin. Let point P be on the
circle and form an angle of θ (in
radians) with the positive x axis.
The cosine θ is the abscissa
of point p, i.e. x = cosine θ.
(0, 1)
P (x, y)
θ
(1, 0)
The sine θ is the ordinate of
point p, i.e. y = sine θ.
Remember the sign “+/-” of the abscissa and the
ordinate in the different quadrants. That will
help you get their signs correct in the future.
6
Trigonometric Functions
The cosine θ is the abscissa of point p, i.e. x = cosine θ.
The sine θ is the ordinate of point p, i.e. y = sine θ.
The sine and cosine of some special angles
Radian
0
π/6
π/4
π/3
π/2
π
2π
Degree
0º
30º
45º
60º
90º
180º
360º
Sine
0
1/2
2/2
3/2
1
0
0
Cosine
1
3/2
2/2
1/2
0
-1
1
The values in aqua will repeat. Know them!
7
Graphs of Sine and Cosine
f (x) = sine x is a periodic
function that repeats every
2π radians and can be
found on your graphing
calculator as:
f (x) = cosine x is a periodic
function that repeats every
2π radians and can be
found on your graphing
calculator as:
8
Graphs of Sine and Cosine
y = sin x
y = cos x
Being able to picture these graphs in my mind has
helped me a lot in determining the numeric value of a
trig function. If you combine this information with the
basic numerical information given earlier you will be in
pretty good shape for getting the correct numerical
values for the trig functions.
9
Four Other Trig Functions
Four Other Trigonometric Functions
sin x
tan x 
cos x
cos x  0
cos x
cot x 
sin x
sin x  0
These functions may also be graphed on your calculator.
10
Four Other Trig Functions
Four Other Trigonometric Functions
1
se c x 
cos x
1
csc x 
sin x
cos x  0
sin x  0
These functions may also be graphed on your calculator.
11
Trigonometric Identities
There are literally hundreds of trig identities.
Several of the most useful follow.
Reciprocal identities
12
Trigonometric Identities
Pythagorean Identities
sin 2 x + cos 2 x = 1
1 + tan 2 x = sec 2 x
1 + cot 2 x = csc 2 x
Quotient Identities
sinx
tanx 
cos x
cos x
cot x 
sin x
13
Trigonometric Identities
Sum Angle Identities
sin(x  y )  sinx cosy  cosx siny
cos(x  y)  cos xcos y
sin xsin y
tan x  tan y
tan (x  y) 
1  tan x tan y
14
Trigonometric Identities
Double Angle Identities
sin(2x)  2sin xcos x
cos(2x)  cos2 x  sin2 x
 2cos x  1
2
 1  2sin2 x
2 tan x
tan (2x) 
1  tan 2 x
15
Trigonometric Identities
And lots more
Handout
16
Summary.
Degree-Radian Conversion Formula
deg
rad


180
 rad
We defined the six trigonometric functions.
We examined the graphs of the trigonometric
functions.
We examined some trig identities.
We are now ready to continue our study of
calculus using the trigonometric functions.
17
ASSIGNMENT
§1.6; Page 28; 1 – 11.
18