Upload
carter-fernandez
View
36
Download
1
Embed Size (px)
DESCRIPTION
1.6 Trig Functions. 1.6 Trig Functions. The Mean Streak, Cedar Point Amusement Park, Sandusky, OH. P. positive angle. x. O. x. O. negative angle. P. Trigonometry Review. (I) Introduction. - PowerPoint PPT Presentation
Citation preview
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.
If OP is rotated counter-clockwisefrom the x-axis, the angle so formed is positive.
But if OP is rotated clockwisefrom the x-axis, the angle so formed is negative.
O
P
xnegative angle
P
O xpositive angle
(II) Degrees & Radians
Angles are measured in degrees or radians.
rr
r1c
Given a circle with radius r, the angle subtended by an arc of length r measures 1 radian.
Care with calculator! Make sure your calculator is set to radians when you are making radian calculations.
180rad
(III) Definition of trigonometric ratios
r
y
hyp
oppsin
r
x
hyp
adjcos
x
y
adj
opptan
cos
sin
sin
1 cosec
cos
1sec
sin
cos
tan
1cot
x
y P(x, y)
r y
x
Note:
1sin
sin
1
Do not write cos1tan1
From the above definitions, the signs of sin , cos & tan in different quadrants can be obtained. These are represented in the following diagram:
All +ve sin +ve
tan +ve
1st2nd
3rd 4th
cos +ve
What are special angles?
(IV) Trigonometrical ratios of special angles
Trigonometrical ratios of these angles are worth exploring
,...,2
,3
,4
30o, 45o, 60o, 90o, …
1
00sin
0sin
12
sin
02sin
12
3sin
sin 0° 0
sin 360° 0sin 180° 0
sin 90° 1 sin 270° 1
xy sin
0 2
23
1
10cos 1cos
02
cos
12cos
02
3cos
cos 0° 1
cos 360° 1
cos 180° 1
cos 90° 0cos 270°
1xy cos
0 2
23
1
00tan 0tan
undefined. is 2
tan
02tan
undefined. is 2
3tan
tan 180° 0
tan 0° 0
tan 90° is undefined tan 270° is undefined
tan 360° 0
xy tan
0 2
23
Using the equilateral triangle (of side length 2 units) shown on the right, the following exact values can be found.
2
3
3sin60sin
2
3
6cos30cos
2
1
6sin30sin
2
1
3cos60cos
33
tan60tan
3
1
6tan30tan
2
2
2
1
4sin45sin
2
2
2
1
4cos45cos
14
tan45tan
Complete the table. What do you observe?
2nd quadrant sin)sin(
cos)cos(
tan)tan(
Important properties:Important properties:
3rd quadrant sin)sin(
cos)cos(
tan)tan(
1st quadrant sin)2sin(
cos)2cos(
tan)2tan(
or 2
Important properties:Important properties:
4th quadrant
sin)2sin( cos)2cos(
tan)2tan(or
or 2
sin)sin( cos)cos(
tan)tan(In the diagram, is acute. However, these relationships are true for all sizes of
Complementary angles
E.g.: 30° & 60° are complementary angles.
Two angles that sum up to 90° or radians are called complementary angles.
2
2
and are complementary angles.
Recall:
2
160cos30sin
2
3
6cos
3sin
3
160cot30tan 330cot60tan
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 . Solve for θ if 2
1)
2
3( 0
4
Basic angle, α =
Since sin is positive, it is in the 1st or 2nd quadrant )2
3(
42
3
42
3 or
Therefore
4
3)(
4
5 orleinadmissib
Hence, 4
3
ry
xA
O
P(x, y)By Pythagoras’ Theorem,
222 ryx
122
r
y
r
x
(VI) 3 Important Identities
sin2 A cos2 A 1
r
xA cos
r
yA sinSince and ,
1cossin 22 AA Note:
sin 2 A (sin A)2 cos 2 A (cos A)2
A2cos
1
(1) sin2 A + cos2 A 1
(2) tan2 A +1 sec2 A
(3) 1 + cot2 A csc2 A
tan 2 x = (tan x)2
(VI) 3 Important Identities
Dividing (1) throughout by cos2 A,
Dividing (1) throughout by sin2 A,
2)(sec A
2
cos
1
A
A2sec
(VII) Important Formulae
(1) Compound Angle Formulae
BABABA sincoscossin)sin( BABABA sincoscossin)sin(
BABABA sinsincoscos)cos( BABABA sinsincoscos)cos(
BA
BABA
tantan1
tantan)tan(
BA
BABA
tantan1
tantan)tan(
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,
tan31
3tan5
3tantan155
8
1tan
A
AA
tantan1
tantan)tan(
(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)
A
AA
2tan1
tan22tan
Proof:
)sin(
2sin
AA
A
AAAA sincoscossin
AAcossin2
)cos(2cos AAA
AA 22 sincos
)cos1(cos 22 AA
1cos2 2 A
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 mode to radians and use when you need to use degrees.
2nd o
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.
y a f b x c d
Vertical stretch or shrink;reflection about x-axis
Horizontal stretch or shrink;reflection about y-axis
Horizontal shift
Vertical shift
Positive c moves left.
Positive d moves up.
The horizontal changes happen in the opposite direction to what you might expect.
is a stretch.1a
is a shrink.1b
When we apply these rules to sine and cosine, we use some different terms.
2sinf x A x C D
B
Horizontal shift
Vertical shift
is the amplitude.A
is the period.B
A
B
C
D 21.5sin 1 2
4y x
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.
Time: .00108 .00198 .00289 .00379 .00471 Pressure: .200 .771 -.309 .480 .581
.00108,.00198,.00289,.00379,.00471 L1 ENTER
2nd { .00108,.00198,.00289,.00379,.00471 2nd }
STO alpha L 1 ENTER
.2,.771, .309,.48,.581 L2 ENTER
SinReg L1, L2 ENTER
2nd MATH 6 3
Statistics Regressions
9
SinReg
alpha L 1 alpha L 2 ENTER
DoneThe calculator should return:
,
Tuning Fork Data
ShowStat ENTER
2nd MATH 6 8
Statistics ShowStat
ENTER
The calculator gives you an equation and constants: siny a b x c d
.608
2480
2.779
.268
a
b
c
d
2nd MATH 6 3
Statistics Regressions
9
SinReg
alpha L 1 alpha L 2 ENTER
DoneThe calculator should return:
,
ExpReg L1, L2 ENTER
We can use the calculator to plot the new curve along with the original points:
Y= y1=regeq(x)
2nd VAR-LINK regeq
x )
Plot 1 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.
2 3
2
2
2
3
2
2
Trig functions are not one-to-one.
However, the domain can be restricted for trig functions to make them one-to-one.
These restricted trig functions have inverses.
Inverse trig functions and their restricted domains and ranges are defined on page 47.
siny x