Transcript PPT

1.6 Trig Functions
1.6 Trig Functions
The Mean Streak, Cedar Point Amusement Park, Sandusky, OH
Trigonometry Review
(I)
Introduction
By convention, angles are measured from the initial line
or the x-axis with respect to the origin.
P
If OP is rotated counter-clockwise
positive angle
from the x-axis, the angle so formed
x
O
is positive.
But if OP is rotated clockwise
from the x-axis, the angle so
formed is negative.
O
x
negative angle
P
(II)
Degrees & Radians
Angles are measured in degrees or radians.
Given a circle with radius r, the
angle subtended by an arc of length r
measures 1 radian.
r
1
c
r
 rad  180
Care with calculator! Make sure your
calculator is set to radians when you are making
radian calculations.
r
(III) Definition of trigonometric ratios
y
P(x, y)
r
y
sin 

sin  
cos  
tan  
hyp
adj
hyp
opp
adj

1
 sin 
x
x
opp
1
Note:



y
cosec  
r
x
r
y
x
sec  

sin 
cos 
1
sin 
1 Do not write
1
1
cos , tan  .
cos 
1
cos 
cot  

tan  sin 
From the above definitions, the signs of sin , cos 
& tan  in different quadrants can be obtained.
These are represented in the following diagram:
sin +ve
2nd
3rd
tan +ve
All +ve
1st
4th
cos +ve
(IV) Trigonometrical ratios of special angles
What are special angles?
30o, 45o, 60o, 90o, …
  
, , , ,...
4 3 2
Trigonometrical ratios of these angles are
worth exploring
y  sin x
1
0

2
1
sin 0  0

3
2
sin 2  0
sin   0

sin 0°  0 sin  1
2
sin 180°  0
sin 90°  1
2
sin 360°  0
3
sin  1
2
sin 270°  1
1
y  cos x
0

2

1
cos 0°  1
cos 2  1
cos 360°  1
cos   1
cos 0  1
2
3
2
cos 180°  1

cos  0
2
cos 90°  0
3
cos  0
2
cos 270°  0
y  tan x
0
tan 0  0
tan 0°  0

2

3
2
tan   0
tan 180°  0
2
tan 2  0
tan 360°  0

tan is undefined.
2
3
tan is undefined.
2
tan 90° is undefined
tan 270° is undefined
Using the equilateral triangle
(of side length 2 units) shown
on the right, the following exact
values can be found.
 1
sin 30  sin 
6 2

3

sin 60  sin 
3 2
 1

cos 60  cos 
3 2

3

cos 30  cos 
6 2

 1
tan 30  tan 
6
3


tan 60  tan  3
3


1
2
sin 45  sin


4
2
2

cos 45  cos

4

1
2

2
2

tan 45  tan  1
4

Complete the table. What do you observe?
Important properties:
2nd quadrant
sin(  )  sin 
1st quadrant
sin(2  )  sin 
cos(   )   cos 
cos( 2  )  cos 
tan(  )   tan 
tan(2  )  tan 
3rd quadrant
sin(  )   sin 
cos(   )   cos 
tan(  )  tan 
Important properties:
4th quadrant
sin(2  )   sin 
cos( 2  )  cos 
tan(2  )   tan 
sin()   sin 
cos( )  cos 
tan()   tan 
In the diagram,  is acute.
However, these
relationships are true for
all sizes of .
Complementary angles
Two angles that sum up to 90° or  radians are called
2
complementary angles.
E.g.: 30° & 60° are complementary angles.
 
 and     are complementary angles.
2

Recall:
1
sin 30  cos 60 
2


1
tan 30  cot 60 
3




3
sin  cos 
3
6 2


tan 60  cot 30  3
Principal Angle & Principal Range
Example: sinθ = 0.5


2

2
Principal range
Restricting y= sinθ inside the principal range makes it a
one-one function, i.e. so that a unique θ= sin-1y exists
Example: sin
Since
(
3
1
) 
2
2
3
sin (   )

2
is positive, it is in the 1st or 2nd quadrant
Basic angle, α = 4
3

 
4
Therefore 2

5
(inadmissib le )
4
Hence,  
3
4
. Solve for θ if 0    
or
3

   
2
4
or

3
4
(VI) 3 Important Identities
P(x, y)
By Pythagoras’ Theorem,
x2  y 2  r 2
2
2
 x  y
     1
r r
y
x
Since sin A 
and cos A  ,
r
r
sin A2  cos A2  1
sin2 A  cos2 A  1
O
r
y
A
x
Note:
sin 2 A  (sin A)2
cos 2 A  (cos A)2
(VI) 3 Important Identities
(1)
sin2 A + cos2 A  1
Dividing (1) throughout by cos2 A,
tan 2 x = (tan x)2
(2)
tan2 A +1  sec2 A
1
Dividing (1) throughout by sin2 A,
(3)
1+
cot2 A

csc2 A
cos 2 A
 1 


 cos A 
 (sec A)
2
 sec A
2
2
(VII) Important Formulae
(1)
Compound Angle Formulae
sin( A  B)  sin A cos B  cos A sin B
sin( A  B)  sin A cos B  cos A sin B
cos( A  B)  cos A cos B  sin A sin B
cos( A  B)  cos A cos B  sin A sin B
tan A  tan B
tan( A  B) 
1  tan A tan B
tan A  tan B
tan( A  B ) 
1  tan A tan B
E.g. 4: It is given that tan A = 3. Find, without using calculator,
(i) the exact value of tan , given that tan ( + A) = 5;
(ii) the exact value of tan  , given that sin ( + A) = 2 cos ( – A)
Solution:
(i)
Given tan ( + A)  5 and tan A  3,
tan   tan A
tan(  A) 
1  tan  tan A
tan   3
5
1  3 tan 
5  15 tan   tan   3
1
 tan  
8
(2)
Double Angle Formulae
(i) sin 2A = 2 sin A cos A
(ii) cos 2A = cos2 A – sin2 A
= 2 cos2 A – 1
= 1 – 2 sin2 A
(iii) tan 2 A 
2 tan A
2
1  tan A
Proof:
sin 2 A
 sin( A  A)
 sin A cos A  cos A sin A
 2 sin Acos A
cos 2 A  cos( A  A)
 cos 2 A  sin 2 A
2
2
 cos A  (1  cos A)
 2 cos 2 A  1
Trigonometric functions are used extensively in calculus.
When you use trig functions in calculus, you must use radian
measure for the angles. The best plan is to set the calculator
o when you need to use
mode to radians and use 2nd
degrees.
If you want to brush up on trig functions, they are graphed
on page 41.

Even and Odd Trig Functions:
“Even” functions behave like polynomials with even
exponents, in that when you change the sign of x, the y
value doesn’t change.
Cosine is an even function because: cos     cos  
Secant is also an even function, because it is the reciprocal
of cosine.
Even functions are symmetric about the y - axis.

Even and Odd Trig Functions:
“Odd” functions behave like polynomials with odd
exponents, in that when you change the sign of x, the
sign of the y value also changes.
Sine is an odd function because:
sin      sin  
Cosecant, tangent and cotangent are also odd, because
their formulas contain the sine function.
Odd functions have origin symmetry.

The rules for shifting, stretching, shrinking, and reflecting the
graph of a function apply to trigonometric functions.
Vertical stretch or shrink;
reflection about x-axis
a  1 is a stretch.
Vertical shift
Positive d moves up.
y  a f b  x  c   d
Horizontal shift
Horizontal stretch or shrink;
Positive c moves left.
reflection about y-axis
b  1 is a shrink. The horizontal changes happen
in the opposite direction to what
you might expect.

When we apply these rules to sine and cosine, we use some
different terms.
A is the amplitude.
Vertical shift
 2

f  x   A sin   x  C    D
B

Horizontal shift
B is the period.
B
4
A
3
C
2
D
1
-1
0
-1
 2

y  1.5sin   x  1   2
 4

1
2
x
3
4
5

The sine equation is built into the TI-89 as a
sinusoidal regression equation.
For practice, we will find the sinusoidal equation for the
tuning fork data on page 45. To save time, we will use only
five points instead of all the data.

Tuning Fork Data
Time:
Pressure:
.00108
.200
.00198 .00289
.771
-.309
.00379
.480
.00108,.00198,.00289,.00379,.00471  L1
2nd
ENTER
{ .00108,.00198,.00289,.00379,.00471
STO
.2,.771, .309,.48,.581  L2
.00471
.581
alpha
}
2nd
L 1
ENTER
ENTER
SinReg L1, L2 ENTER
2nd
MATH
6
Statistics
3
9
alpha
SinReg
Regressions
L 1
,
alpha
The calculator
should return:
L 2
Done
ENTER

ExpReg L1, L2 ENTER
2nd
MATH
6
Statistics
3
9
alpha
SinReg
Regressions
L 1
,
alpha
The calculator
should return:
L 2
ENTER
Done
ShowStat ENTER
2nd
MATH
6
Statistics
8
ENTER
ShowStat
a  .608
The calculator gives
you an equation and
y  a sin  b  x  c   d
constants:
b  2480
c  2.779
d  .268

We can use the calculator to plot the new curve along with
the original points:
Y=
2nd
Plot 1
y1=regeq(x)
VAR-LINK
x
)
regeq
ENTER
ENTER
WINDOW

Plot 1
ENTER
ENTER
WINDOW
GRAPH

WINDOW
GRAPH
You could use the
“trace” function to
investigate the pressure
at any given time.

Trig functions are not one-to-one.
However, the domain can be restricted for trig functions
to make them one-to-one.

2 
y  sin x
3
2


2

2
3
2

2
These restricted trig functions have inverses.
Inverse trig functions and their restricted domains and
ranges are defined on page 47.
