**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