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Chapter 4: Trigonometric Functions

Precalc 7th hour
by

Vandhana Murali

on 17 January 2014

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Transcript of Chapter 4: Trigonometric Functions

Chapter 4: Trigonometric Functions
Inverse Trigonometric Functions
Inverse Sine Function





Inverse Cosine Function





Inverse Tangent Function





Inverse Properties
Applications and Examples
Example 1
Domain: (- , )
Range: [-1, 1]
Amplitude: 1
Period: 2pi or 360°
Example 2
Jokes
4.1 - Summary
Identities
SOHCAHTOA
Basic Right Triangle Identities
y= cos(x)
Angle of Elevation/Depression
Graphs of Sine and Cosine Functions
By: Alina, Jaynee, Isabel, & Vandhana
Domain: All real numbers
Range: [-1, 1]
Amplitude: 1
Period: 2pi or 360°
Endpoints
y = sin(x)
4.5
4.7
Defined by y=arcsinx if and only if siny=x
Domain: [ -1, 1 ]
Range: [ -pi/2, pi/2 ]
Used to find the angle in degrees of which the sine value is x
Defined by y=arccosx if and only if cosy=x
Domain: [ -1, 1 ]
Range: [ 0, pi ]
Used to find the angle in degrees of which the cosine value is x
Defined by y=arctanx if and only if tany=x
Domain: ( -infinity, infinity )
Range: ( -pi/2, pi/2 )
Used to find the angle in degrees of which the tangent value is x
Reminders
General Equation of a sinusoidal graph: y = asin(bx - c) + d

Amplitude: half the distance between the maximum and minimum values of the function

y = asin(x) y = acos(x) Amplitude = |a|

Period: cycle length of a curve on a sinusoidal graph

Period = original period / b

Horizontal/Phase Shift: c/b
Vertical Shift: d
Use these equations to find the left and right endpoints of a one-cycle interval.
bx - c = 0
bx - c = original period
y = 2 + 3cos(2x)
Amplitude: 3
Period: 2pi/b = 2pi/2 = pi

Draw sinusoidal axis: y = 2
Get 3 key points to graph
the function
When x = 0, y = 5
y = 2 + 3cos(2*0)
= 2 + 3cos(0)
= 2 + 3(1)
= 5
Since this is a cosine function and the period is "pi", we know that when x = 3.14, y = 5.
Another key point would be when
x = pi/2 . Since it is the minimum point we know "y" should be 5 units below the sinusoidal axis. Which, in this case, is -1.
y = 4 - cos(x + (pi/2))
Amplitude: |- 1 | = 1 (since "a" is negative, the graph of cos(x) is flipped.)
Period: 2pi
Sinusoidal axis: 4
Phase shift: -pi/2 (c = -pi/2)
Endpoints:
x - (-pi/2) = 0 x = -pi/2
x - (-pi/2) = 2pi x = 3pi/2
1. Draw sinusoidal axis
2. Find some points
Use amplitude to get the 3 points ( you can find the maximums and minimum.
Or you can use the function
by plugging in different values for x. - this takes more time though.
4.6
y = tan(x)

Period: pi
Domain:
Range:(- , )

Graphs of Other Trigonometric Functions
y = cot(x)
Period: pi
Domain:
Range: (- , )
Other Functions
y = csc(x)
Period: 2pi
Domain:
Range: (- , -1], [1, )
Symmetry: Origin
y = sec(x)
Period: 2pi
Domain: (- , -1], [1, )
Range:
Symmetry: y-axis
Example 1
y = 1 + 2sec(4x)
Amplitude: 2
Period: 2pi/4 = pi/2
Vertical shift = 1
1. Draw sinusoidal axis
2. Using the amplitude
and the graph of sec(x)
we can find the 3 key points.
Example 2
y = (1/4)cot(x + (pi/2))
Amplitude: 1/4
Period: pi
Phase Shift: -pi/2
Endpoints:
x - (-pi/2) = 0 x = -pi/2
x - (-pi/2) = pi x = pi/2
Graph using amplitude and cot(x)
New domain (because of phase shift)
So at the endpoints, the
graph will be undefined.
Graph of cot(x) and (1/4)cot(x)
(without phase shift)
y = (1/4)cot(x + (pi/2))
Quotient
4.3

Right Triangle Trig
Reciprocal
Pythagorean
Example #1
A person is standing 50 ft. from the base of a pine tree. The person measures the angle of elevation to the top of the tree as 72 degrees. How tall is the tree?
Tan 72
=
X
50 ft.
y= 50 tan 72
y=
153.88 ft.
X
angle of elevation=72 degrees
50 ft.
Example #2
8
15
Find the exact values of all 6 trigonometric functions of the angle
sin
SOHCAHTOA!
= 8/17
cos
tan
csc
sec
cot
=15/17
=8/15
=17/8
=17/15
=15/8
if theta = 35 degrees, what is the value of y?
y
sin35 =
8
y
y =
sin35
8
= 13.95
4.4
Reference angle: acute angle theta formed by the terminal side of theta and the horizontal axis
theta = A in the image
To find the value of a trigonometric function of any angle theta:
Determine the function value for the associated reference angle
1
2
Depending on which quadrant theta lies, prefix the appropriate sign to the function value
Finding the Trig function of any angle
(x,y)
r
theta
x
y
angle theta with (x,y) on a point on the terminal side of theta and
r=
=
0
sin
=
y
r
cos
=
x
r
tan
=
y
x
sec
=
r
x
csc
=
r
y
cot
=
x
y
The denominators of tangent, cotangent, secant and cosecant cannot equal zero!
Example #1
Evaluate the trigonometric function:
cos
4
3
lies in Quadrant lll
reference angle, theta = (4
/3)-
=
/3
remember it is negative!
= -1/2
so, using the
(-1/2,
3
/2)
Tip: Know your unit circle well!
Example #2
Find the reference angle theta and sketch a picture of theta in standard position
y
x
theta
theta = 208 degrees
reference angle
reference angle= 28 degrees
208 degrees - 180 degrees
= 28 degrees
sin(arcsinx)=x and arcsin(siny)=y
cos(arccosx)=x and arccos(cosy)=y
tan(arctanx)=x and arctan(tany)=y
60 cm
A
100 cm
Find angle A.
sinA=60/100
arcsin(sinA)=arcsin(60/100)
A=arcsin(60/100)
A=35.87 degrees
X
11 cm
17 cm
Find angle X.
tanX=17/11
arctan(tanX)=arctan(17/11)
X=arctan(17/11)
X=57.09 degrees
4.8
The shadow of a lamppost is 13 feet long and angle Y is 50 degrees. How far is it from the end of the shadow to the top of the lamppost?
13 feet
Y
X
cos50=13/X
13/(cos50)=X
X=20.22 feet
A boat leaves the port heading in the direction N15W. After 6 miles, the boat turns to N70W and travels three more miles. How far is the boat from the port?
6 mi
15
70
3 mi
X
3 mi
20
1.03 mi
Use the sine and cosine functions to find the remaining sides.
2.82 mi
75
6 mi
5.79
mi
6.82
mi
4.37 mi
Use addition and the Pythagorean Theorem.
The boat is 8.10 miles from the port.
Three statisticians go out hunting together. After a while they spot a solitary rabbit. The first statistician aims and overshoots. The second aims and undershoots. The third shouts out "We got him!"
You learn how to:
* Estimate angle measurement
* Determine what quadrant angles are in
* Sketch angles in standard form
* Co-terminal angles
* Find complement and supplement angles
* Change angle measure from radians to degrees and back(with and without a calculator)
* Angular and linear velocity
4.1 Definitions/Theorems
1) Trigonometry: Measurement of triangles
2) Angle: A figure formed by two rays
3) Initial Side: The starting position of the ray
4) Terminal Side: The position after rotation
5) Vertex: The endpoint of the ray of an angle
6) Standard Position: When the vertex of an angle is the origin and the initial side coincides with the positive x-axis
7) Positive Angles: Created by counter-clockwise rotation
8) Negative Angles: Created by clockwise rotation
9) Coterminal Angles: Angles with the same initial and terminal sides
10) Central Angle: An angle with a vertex in the center of the circle
11) Radian: The radius of a circle
12) Complementary Angles: Angles whose sum is (pi/2)
13) Supplementary Angles: Angles whose sum is pi

Conversion between degree radians:
* degrees x pi radians/180 degrees --> degrees to radians
* radians x 180 degrees/pi radians --> radians to degrees

Linear and Angular Speed:
*Linear: The speed at which the outside tip of the radius is traveling
* arc length/time = s/t
* Angular: The speed at which something rotates
* central angle/time = theta/t



To sum it all up...
Hope that didn't subtract from our presentation!

You learn how to:
* Determine the value of trig. functions
* Find a point (x,y) on the unit circle that corresponds to a real number
* Evaluate the sine, cosine and tangent of a real number
* Evaluate trig. functions of a real number
* Sketch rational function graphs
4.2 Summary


1) Unit Circle: A circle with a radius of one
2) Trig. Functions: Sine = o/h, Cosine = a/h, Tangent = o/a, Cosecant = 1/sine, Cotangent = cos/sin, Secant = 1/cos
3) Periodic: Functions that behave in a repetitive manner

Definition of a periodic function:
A function (f) is periodic i there exists a positive real number (c)
* f(t+c) = f(t)

Even/Odd Trig. Functions:
* Cosine/Secant functions are even
Cos(-t) = cos t
Sec(-t) = sec t
* Sine, Cosecant, tangent and cotangent functions are odd
Sin(-t) = -sin t
Tan(-t) = -tan t
Csc(-t) = -csc t
Cot(-t) = -cot t

4.1 Examples
4.2 Examples
4.1
4.2
8.10 mi
4.2 Definitions/theorems
1.55 mi
Full transcript