§1.1

How is an angle formed?

§1.1

What is the vertex?

§1.1

What are the sides of an angle also called?

§1.1

What is the vertex of the angle(Θ) labled?

[image]

The vertex of the angle Θ is labled B.

Also denoted as ABC, where

the letter associated with the vertex

is written between

the letters associated with

the points on each side.

§1.1

Which side, (BC) or (BA), is the initial side of the angleΘ?

[image]

§1.1

Which side, (BC) or (BA), is the terminal side of the angleΘ?

 [image]

§1.1

When is the angle formed considered a positive angle?

When the rotation from the initial side

to the terminal side

takes place in a

counterclockwise direction.

§1.1

When is the angle formed considered a negative angle?

When the rotation formed

from the initial side

to the terminal side

takes place in a

clockwise direction.

§1.1

How many degrees are in an angle formed by

one complete revolution?

§1.1

What is the fraction of 1° out of a full rotation?

1° = _1_

        360

§1.1

What are angles called, that measure 90°?

§1.1

What are angles called, that measure 180°?

§1.1

What are angles called, that measure between 0° and 90°?

§1.1

What are angles called, that measure between 90° and 180°?

§1.1

What are two angles, the sum of the measure of which is 90°?

§1.1

What is the measure of the compliment of an angle with a measure of 60°?

The compliment of an agle, with the measure of 60°, has a measure of 30°.

Because:

60° + 30° = 90°

§1.1

What are two angles, the sum of the measure of which is 180°?

§1.1

What is the measure of the supplement of an angle with a measure of 45°?

The supplement of an angle with a measure of 45°, has a measure of 135°.

Because:

45° + 135° = 180°

§1.1

"Angle Θ" can also mean what?

§1.1

What is an equilateral triangle?

§1.1

What is an isosceles triangle?

§1.1

What is a scalene triangle?

§1.1

What is an acute triangle?

§1.1

What is an obtuse triangle?

§1.1

What is a right triangle?

§1.1

What is the hypotenuse?

§1.1

What are the two sides of a right triangle that are not the hypotenuse?

§1.1

The sum of angles in any triangle are.... ?

§1.1

In a right triangle, what are the two angles considered, which are not right angles?

§1.1

According to the Pythagorean Theorem:

"In any right triangle, the square of the length of the longest side (the hypotenuse) is equal to..." what?

"...is equal to the sum of the squares of the lengths of the other two sides (legs)."

[image]

If "C" = 90°, then  c²=a²+b²

§1.1

What is the formula for the area of a triangle?

§1.1

What is a Pythagorean triple?

Whenever the three sides in a right triangle are natural numbers. 

After "3,4,5":

[image]

§1.1

What are the relative measures of a 30°-60°-90° triangle?

[image]

In any right triangle in which the two acute angles are 30° and 60°;

*the hypotenuse is always twice the shortest side (a),

*and the side of medium length (opposite the 60° angle) is always √3 times the shortest side.

§1.1

What are the relative measures of a 45°-45°-90° triangle?

[image]

If the two acute angles in a right triangle are both 45°;

*the two shorter sides (legs) are equal

*and the longest side (hypotenuse) is √2 times as long as the shorter sides.

That is, if the legs are of length n, then the hypotenuse has length n√2.

§1.2

 Who invented the rectangular coordinate system?

§1.2

What four sections is the cartesian coordinate system divided into by the axes?

Quadrants

[image]

§1.2

What is the idea of the rectangular coordinate system?

It can measure position on a plane.

Ex: the point (-2,4) is plotted below;

[image]

§1.2

What is the definition of a graph?

A picture of all solutions to an equation in 2 variables. 

Ex; The equation of Y=2X-1 is graphed as;

[image]

§1.2

Whatis the output, or dependent, of the equation Y=2X?

§1.2

Whatis the input, or independent, of the equation Y=2X?

§1.2

How can a slope be verified on a graphing calculator?

To verify any point (other than the origin) on the line Y=(3/2)X ; 

1. Set window to -6≤X≤6 and -6≤Y≤6 with [graph], [F2{wind}], reset; "Xmin=-6, Xmax=6, Ymin=-6, Ymax=6" then [F1y(x)=]
2.  Define function as "Y1=3[x-var]/2" [2nd][F4{trace}]
3. Slide tracer along graph to check points of solution and verify slope ratio.

§1.2

Any parabola that opens up or down can be described by an equation of what form?

§1.2

In an equation of the form Y = a (X - h)² + k , which two variables indicate the point of the vertex?

§1.2

In an equation of the form Y = a (X - h)² + k , how does the variable "a" change the shape of the graph of the parabola?

The shape of:

+a = upward facing

-a = downward facing

a>1 = narrow

a<1 = wide

[image]

§1.2

How can the equation of a parabola be verified on a graphing calculator?

To verify the parabola of Y=-(7/640)(X-80)²+70:

1. Set window settings to 0≤X≤180, scale=20; 0≤Y≤80, scale 10
2. Define function as Y1=-(7/640)(X-80)²+70, [2nd][F5{graph}]
3. Evaluate [more][more][F1{EVAL}] at "0" [ENTER].  Blinking point? check!  Then evaluate again [EXIT][F1{EVAL}] at "160 [ENTER].  Blinking point? check!

§1.2

The distance between any two points (X₁,Y₁) and (X₂,Y₂) are given by what formula?

The Distance Formula:

[image]

§1.2

How can the distance formula be derived?

By applying the Pythagorean Theorem to a right triangle. 

Let the hypotenuse = r

r²= (X₂-X₁)²+(Y₂-Y₁)²

r=√{(X₂-X₁)²+(Y₂-Y₁)²}

[image]

§1.2

What is the definition of a circle?

§1.2

What is the radius of a circle?

§1.2

What is the center of a circle?

§1.2

Apply the distance formula to a circle if:

r>0 = radius

(h-k) = the center

(x,y) = any point on the circle

§1.2

What is the equation of a circle?

§1.2

What is a unit circle?

When the radius = 1

x²+Y²=1

§1.2

How is the circle x²+y²=1 graphed on a calculator?

1. Isolate Y; y²=1-x²,  y=±√{1-x²}
2. Define each resulting function seperately as "Y1 = √(1-X²)"  and  "Y2 = -√(1-X²)
3. Set window and scale as needed

§1.2

What does ± mean when found in the function of a circle ( y=±√{1-x²} ) ?

The circle is the union of two seperate functions:

+√... represents the top half of the circle.

-√... represents the bottom half of the circle.

§1.2

When is an angle in standard form/position?

If its initial side is along the positive X-axis and its vertex is at the origin. 

The angle itself is positive, meaning counterclockwise.

§1.2

What does "Θ∈QI" mean?

If angle Θ is in standard position and the terminal side of Θ lies in Quadrant I, then we say

"Θ lies in quadrant I"

abbreviated thusly;

"Θ∈QI"

§1.2

What is a quadrant angle?

The terminal side of the angle, in standard position, lies along one of the axes.

[image]

§1.2

What are coterminal angles?

Two angles in standard position with the same terminal side.

[image]

§1.2

If (k) is any integer, what is the formula for any angle that is coterminal (c) with (-90)°?

§1.3

Definition I

If Θ is an angle in standard position,

and the point (x,y) is

any point on the terminal side of Θ

other than the origin, then

the six trigonometric functions of angle Θ are:

The sine of Θ

The cosine of Θ

The tangent of Θ

The cotangent of Θ

The secant of Θ

The cosecant of Θ

(each defined on seperate notecards)

§1.3

sinΘ

(According to definition I)

y

r

§1.3

cosΘ

(According to definition I)

x

r

§1.3

tanΘ

(According to definition I)

y

x

(x≠0)

§1.3

cotΘ

(According to definition I)

x

y

(y≠0)

§1.3

secΘ

(According to definition I)

r

x

(x≠0)

§1.3

cscΘ

(According to definition I)

r

y

(y≠0)

§1.3

What is "r" in Definition I?

"r" is the distance from the origin to (x,y) where:

x²+y²=r²

or

r=√(x²+y²)

§1.3

What are the algebraic signs, + or -, of the six trigonomic functions in Quadrant I?

All positive:

sinΘ and cscΘ +

cosΘ and secΘ +

tanΘ and cotΘ +

§1.3

What are the algebraic signs, + or -, of the six trigonomic functions in Quadrant II?

sinΘ and cscΘ +

cosΘ and secΘ -

tanΘ and cotΘ -

 §1.3

   What are the algebraic signs, + or -, of the six trigonomic functions in Quadrant III?

sinΘ and cscΘ -

cosΘ and secΘ -

 tanΘ and cotΘ +

 §1.3

   What are the algebraic signs, + or -, of the six trigonomic functions in Quadrant IV?

sinΘ and cscΘ -

cosΘ and secΘ +

 tanΘ and cotΘ -

§1.4

What is the reciprocal identity of cscΘ?

 cscΘ=_1_

           sinΘ

§1.4

What is the reciprocal identity of secΘ?

secΘ=_1_

          cosΘ

§1.4

What is the reciprocal identity of cotΘ?

cotΘ=_1_

          tanΘ

§1.4

What is the equivalent form of

the reciprocal identity of cscΘ=_1_?

                                           sinΘ

sinΘ=_1_

         cscΘ

§1.4

What is the equivalent form of

the reciprocal identity of secΘ=_1_?

                                           cosΘ

cosΘ=_1_

           secΘ

§1.4

What is the equivalent form of

the reciprocal identity of cotΘ=_1_?

                                          tanΘ

tanΘ=_1_

          cotΘ

§1.4

What is the ratio identity of tanΘ?

(and why?)

tanΘ = sinΘ

           cosΘ

because

sinΘ  =  y/r  =  y  =  tanΘ

cosΘ      x/r      x             

§1.4

What is the ratio identity of cotΘ?

(and why?)

cotΘ = cosΘ

           sinΘ

because

cosΘ  =  x/r  =  x  =  cotΘ

sinΘ      y/r      y            

§1.4

What does the notation of sin²Θ mean?

The notation of sin²Θ

is shorthand notation for (sinΘ)². 

It indicates we are to square the number

that is the sine of Θ.

§1.4

What is the first Pythagorean Identity?

§1.4

What are the two equivalent forms

of the first Pythagorean Identity?

cosΘ =±√(1 - sin²Θ)

and

sinΘ =±√(1 - cos²Θ)

§1.4

What is the second Pythagorean Identity?

1 + tan²Θ = sec²Θ

§1.4

What is the third Pythagorean Identity?

1 + cot²Θ = csc²Θ

§1.5

What does it mean to write

something "in terms of" something else?

ex. Write tanΘ "in terms of" sinΘ.

To write tanΘ "in terms of" sinΘ means

to write an expression

that is equivalent to tanΘ

but involves no trigonometric function

other than sinΘ.

§2.1

What is a topographic map?

A type of mapcharacterized by large-scaledetail and quantitative representation of relief, usually using contour lines in modern mapping.

[image]

§2.1

What are contour lines?

The curved lines on a topographic map; they are used to show the changes in elevation of the land shown on the map.

[image]

§2.1

What is the etymological definition of Trigonometry?

It is derived from two Greek words;

1. tri'gonon: meaning triangle.
2. met'ron: meaning measure.

§2.1

Definition II

If triangle ABC is a right triangle with C=90°,

[image]

then the six trigonometric functions for A are defined as follows:

[image]

§2.1

sin A

(According to definition II)

sin A =_side opposite A_= a

              hypotenuse        c

§2.1

cos A

(According to definition II)

cos A =_side adjacent A_= b

              hypotenuse        c

§2.1

tan A

(According to definition II)

tan A =_side opposite A_= a

            side adjacent A     b

§2.1

cot A

(According to definition II)

cot A =_side adjacent A_= b

            side opposite A     a

§2.1

sec A

(According to definition II)

sec A =__hypotenuse__= c

            side adjacent A    b

§2.1

csc A

(According to definition II)

csc A =__hypotenuse__= c

            side opposite A   a

§2.1

What are the cofunctions of Definition II?

Sine and cosine are cofunctions,

as are tangent and cotangent,

and secant and cosecant. 

We say sine is the cofunction of cosine,

and cosine is the cofunction of sine.

Ex:

[image]

sin A =_side opposite A_= a = cos B

    hypotenuse       c

=

cos B =_side adjacent B_= a = sin A

     hypotenuse       c

§2.1

What does "sinus rectus complementi" mean?

The prefix co- in cosine, cosecant, and cotangent is a reference to the complement.

Around the year 1463, Regiomontanus used the term sinus rectus complementi, presumably referring to the cosine as the sine of the complemantary angle. 

In 1620, Edmund Gunter shortened this to co.sinus, which was further abbreviated as cosinus by John Newton in 1658.

§2.1

What is the Cofunction Theorem?

A trigonometric function of an angle is always equal to the cofunction of the compliment angle.

[image]

§2.1

What are the six trigonometric functions

for angle A?

If A = 0°, 90°,

(and from the two special triangles)

30°, 45° and 60°?

§2.1

What is the exact value of sin0°?

0

§2.1

What is the exact value of sin30°?

_1_

2

§2.1

What is the exact value of sin45°?

_1_

√2

§2.1

What is the exact value of sin60°?

_√3_

2

§2.1

What is the exact value of sin90°?

1

§2.1

What is the exact value of cos0°?

1

§2.1

What is the exact value of cos30°?

_√3_

2

§2.1

What is the exact value of cos45°?

_1_

√2

§2.1

What is the exact value of cos60°?

_1_

2

§2.1

What is the exact value of cos90°?

0

§2.1

What is the exact value of tan0°?

0

§2.1

What is the exact value of tan30°?

_1_

√3

§2.1

What is the exact value of tan45°?

1

§2.1

What is the exact value of tan60°?

_√3_

1

§2.1

What is the exact value of tan90°?

∞

§2.1

What is the exact value of cot0°?

∞

§2.1

What is the exact value of cot30°?

√3

1

§2.1

What is the exact value of cot45°?

1

§2.1

What is the exact value of cot60°?

_1_

√3

§2.1

What is the exact value of cot90°?

0

§2.1

What is the exact value of sec0°?

1

§2.1

What is the exact value of sec30°?

_2_

√3

§2.1

What is the exact value of sec45°?

_√2_

1

§2.1

What is the exact value of sec60°?

2

1

§2.1

What is the exact value of sec90°?

∞

§2.1

What is the exact value of csc0°?

§2.1

What is the exact value of csc30°?

2

1

§2.1

What is the exact value of csc45°?

√2

1

§2.1

What is the exact value of csc60°?

_2_

√3

§2.1

What is the exact value of csc90°?

1

§2.2

What fraction is one minute of a degree?

A degree can be divided into 60 minutes. 

1°=60'  or  1'=(1/60)°

§2.2

What is the fraction of one second

of one minute?

A minute can be divided into 60 seconds.

1'=60"  or  1"=(1/60)'

§2.2

How is the expression 13° 24' 15" read?

Thirteen degrees,

Twenty four minutes,

Fifteen seconds.

§2.3

How are significant digits found?

The number of significant digits

(or figures) in a number

is found by counting all the digits

from left to right

beginning with the first nonzero digit

on the left.

When no decimal point is present,

trailing zeros are not considered significant.

§2.3

If the accuracy of sides of a triangle

is to two significant digits,

what is the acuracy of angles?

§2.3

If the accuracy of sides of a triangle

is to three significant digits,

what is the acuracy of angles?

The acuracy of angles is to the

nearest 10 minutes or tenth of a degree.

§2.3

If the accuracy of sides of a triangle

is to four significant digits,

what is the acuracy of angles?

The accuracy of angles is to the

nearest minute or hundredth of a degree.

§2.4

Α     ɑ

§2.4

Β     β

§2.4

Γ     γ(GREEK) or (LATIN)ɣ

§2.4

Δ     δ

§2.4

Ε     ε

§2.4

Θ     θ

§2.4

Σ     σ

§2.4

Ω     ω

§2.4

What is the general formula for

a polygon with "n" sides?

P = _180 ( n - 2 )_

     n

§2.4

What is the definition of

an angle of elevation?

An angle measured from the horizontal up.

[image]

§2.4

What is the definition of

an angle of depression?

An angle measured from the horizontal down.

[image]

§2.4

What is the mathematical definition

of line of sight?

If an observer is positioned

at the vertex of the angle

and views an object in the direction of

the nonhorizontal side of the angle. 

That side is sometimes called

the line of sight of the observer.

[image]

§2.4

What is the bearing of a line?

The bearing of the line l

is the acute angle formed

by the north-south line

and the line l. 

§2.4

What is the notation used to designate the bearing of a line?

The notation used

begins with N (north) or S (south),

followed by the number of degrees in the angle, and ends with E (east) or W (west).

[image]

§2.5

Who discovered vectors?

§2.5

What are vector quantities?

§2.5

What are scalars?

Quantities that have magnitude only,

no direction.

§2.5

What are some vector quantities?

force

velocity

acceleration

§2.5

How is notation used to distinguish between vectors and scalars?

Letters representing vectors in boldface type:

[image]

On paper, put an arrow above letters representing vectors:

[image]

The magnitude of a vector (a scalar) is represented with absolute value symbols:

[image]

§2.5

What is zero vector?

A vector with a magnitude of zero,

and no defined direction.

Denoted by 0.

§2.5

Is the position of a vector in space important?

§2.5

When are two vectors equivalent?

When the two vectors have

the same magnitude and direction.

[image]

§2.5

What is the resultant vector?

The sum of two vectors; A + B

It is the vector that extends

from the tail of A to the tip of B

when the tail of B is placed at the tip of A.

[image]

§2.5

 What is the parallelogram method of adding vectors?

In the parallelogram method for vector addition, the vectors are translated, (moved)

to a common origin, shown on the left:

[image]

§2.5

What is the head to tail method for adding vectors?

Place the two vectors next to each other

such that the head of one vector

is touching the tail of the other vector. 

Shown on the right:

[image]

§2.5

Which end of the vector is the "head"?

The head of the vector is the end with the arrow.  The "pointy" end.

§2.5

Which end of the vector is the "tail"?

§2.5

How are vectors subtracted?

Add the opposite of one vector to the other:

B-A = B+(-A)

[image]

§2.5

What is the tail to tail method of

subtracting vectors?

Put the tails of the two vectors together and then draw the resulting vector from the tip of one to the tip of the other, completing a triangle.

[image]

§2.5

What is the standard position for a vector?

When a coordinate system is superimposed on a vector so that the tail of the vector is at the origin.

[image]

§2.5

What is the horizontal vector component

of the vector A?

A_x:

[image]

§2.5

What is the vertical vector component

of the vector A?

A_y:

[image]

§2.5

What is force?

A push or a pull.

[image]

§2.5

What is static equilibrium?

The state of an object that is stationary (at rest).

When an object is in this state,

the sum of the forces acting on the object

must be equal to the zero vector 0.

§2.5

What is tension?

§2.5

What is work, by mathematical definition?

If a constant force F is applied to an object

and moves the object in a straight line

a distance d in the direction of the force,

(and only in the direction of the force)

then the work W performed by the force is:

W=lFl*d

§2.5

What is a common unit of measure for work?

Foot-pound

"ft-lb"

§3.1

What are sheaves?

A wheel or roller

with a groove along its edge

for holding a belt, rope or cable.

When hung between two supports

and equipped with a belt, rope or cable,

one or more sheaves make up a pulley.

§3.1

What is a reference angle?

For any angleθ in standard position,

the positive acute angle

between the terminal side of θ (0°)

and the x-axis (90°). 

Denoted by [image].

Sometimes called a related angle.

§3.1

Where is [image] if θ ε QI?

§3.1

Where is [image] if θ ε QII?

§3.1

Where is [image] if θ ε QIII?

§3.1

 Where is [image] if θ ε QIV?

§3.1

What is the Reference Angle Theorem?

A trigonometric function of

an angle and its reference angle

are the same,

except, perhaps, for a difference in sign.

§3.1

What are the 4 steps to find

the trigonometric functions of angles

between 0° and 360°?

1. Find [image], the reference angle.
2. Determine the sign of the trigonometric function based on the quadrant in which θ terminates.
3. Write the original trigonometric function of θ in terms of the same trigonometric function of [image].
4. Find the trigonometric function of [image].

§3.1

What are the trigonometric functions of an angle always equal to?

The trigonometric functions of an angleθ

are always equal to

any angle coterminal to θ.

For any integer k, sine and cosine:

sin(θ+360°k) = sinθ

cos(θ+360°k) = cosθ

§3.1

How do we find values of trigonometric functions for an angle larger than 360° or smaller than 0°?

Find an angle between 0° and 360°

that is coterminal to the original angle

and then use the same 4 steps

to find the trigonometric functions.

§3.1

How do we find trigonometric functions of angles that do not lend themselves to exact values?

Find an approximation for sinθ, cosθ, or tanθ

by entering the angle in a calculator

(in degree mode).

§3.2

What is the central angle of a circle?

§3.2

What is a radian?

In a circle, a central angle that cuts off an arc

equal in length to the radius of the circle

has a measure of 1 radian (rad).

[image]

§3.2

What is radian measure?

If a central angle θ, in a circle of radius r,

cuts off an arc of length s, then:

θ(in radians)=s/r

[image]

§3.2

What is the approximate degree measure

of one radian?

§3.2

What is the degree measure of θ in radians?

360° = 2π rad

180° = π rad

§3.2

What is the radian measure of θ in degrees?

_π_ rad = 1°

180             

and

1 rad = (180/π)°

§3.2

How do we convert from degrees to radians?

multiply both sides of the equaion

1°=_π_ rad

180

by the degree value

(θ)1°=(θ)_π_ rad

         180

§3.2

What is 0° in radians?

0 rad

§3.2

What is 30° in radians?

_Π_

6

§3.2

What is 45° in radians?

_Π_

4

§3.2

What is 60° in radians?

_Π_

3

§3.2

What is 90° in radians?

_Π_

2

§3.2

What is 180° in radians?

Π

§3.2

What is 270° in radians?

_3Π_

2

§3.2

What is 360° in radians?

2Π

§3.2

What is the exact value of sin180°?

0

§3.2

What is the exact value of sin270°?

-1

§3.2

What is the exact value of sin360°?

0

§3.2

What is the exact value of cos180°?

-1

§3.2

What is the exact value of cos270°?

0

§3.2

What is the exact value of cos360°?

1

§3.2

What is the exact value of tan180°?

0

§3.2

What is the exact value of tan270°?

undefined

§3.2

What is the exact value of tan360°?

0

§3.3

According to Hipparchus's table of chords, what is a sine function?

The sine of a central angle on the unit circle is half the chord of twice the angle.

chord(θ)= AB = 2AF = 2sin(θ/2)

[image]

§3.3

How can the ordered pair (X,Y) can be written, according to the unit circle?

(cosθ,sinθ)

§3.3

What is the distance from (1,0) to (x,y), according to the unit circle?

If θ is measured in radians;

θ=_t_ = _t_ = t

r        1

The length of the arc t, from (1,0) to (x,y)

is exactly the same as the

radian measure of angle θ.  Therefore:

cosθ = cost = x

sinθ = sint = y

§3.3

What is the Definition (III) for circlular functions?

If (x,y) is any point on the unit circle, and t is the distance from (1,0) to (x,y) along the circumference of the unit circle then:

• cost=x
• sint=y
• tant=y/x (x≠0)
• cott=x/y (y≠0)
• csct=1/y (y≠0)
• sect=1/x (x≠0)

[image]

§3.3

How is the unit circle graphed on a graphing calculator, using Definition III, in radians?

1. Set graphing calculator to parametric mode "param" and radian mode.
2. Define the pair of functions: X=cos, Y=sin_1Tt_1Tt
3. Set window variable to: 0≤t≤2π, scale=π/12, -1.5≤x≤1.5, -1.5≤y≤1.5
4. Graph equation using zoom-square command "zsqr".
5. "Trace" (number of increment)*(π/12) to find the sine and cosine of the increment in radians.

§3.3

How is a unit circle graphed on a graphing calculator, using Definition III, in degrees?

1. Set graphing calculator to parametric mode "param" and degree mode.
2. Set window to 0≤t≤360 with scale=5.
3. Define the pair of functions: X_1T=cost, Y_1T=sint
4. Graph equation using zoom-square command "zsqr".
5. "Trace" (number of increment)*(5)° to find the sine and cosine of the increment in radians.

§3.3

What is a sine function?

y=sinx is identical notation to y=f(x)

X =

• the input
• a real number
• the argument of the function
• a distance along the circumference of the unit circle
• an angle in radians

§3.3

How is Definition III used to find the domain for each of the circular functions?

Because any value t

determines a point (x,y) on the unit circle,

the sine and cosine functions are always defined

and therefore always have a domain of all real numbers.

§3.3

What is the domain for the circular function sint?

The domain for the circular function sint is

All real numbers

(-∞,∞)

§3.3

What is the range of the circular function sint?

[-1,1]

§3.3

How are the values

of the six trigonometric functions,

based on circular definitions,

represented geometrically?

If the point B(x,y) is θ units from point D(1,0), then:[image]

[image]

§3.4

What is the arc length of a circle, cut off by the central angle θ (in radians), if the radius is r?

If θ (in radians) is a central angle in a circle with the radius r, then the length of the arc cut off by θ is given by:

s=rθ

§3.4

If there is an arc length s

between two points A and B,

with a radius of 1800ft

and a central angle of 1°,

what is s?

To find the arc of AB:

1)  We convert θ to radians by multiplying Π/180.

2)  Then apply the formula s=rθ.

s=rθ

rθ=1,800(1)(Π/180)

1,800(1)(Π/180)=10Π

10Π≈31ft

§3.4

Area of the sector is to the area of a circle

as θ is to ____?

"One full rotation."

Area of the sector->A__ = θ_<--Cental angle θ

Area of circle------>Πr²    2Π<----One rotation

(solve for A)

§3.4

What is the area of a sector formed by the central angleθ (in radians) of a circle, if the radius is r?

A =_1_(r²θ)

2  

[image]

§3.5

How are units of linear velocity given?

The units of linear velocity are:

miles per hour

feet per second

distance per time

§3.5

How are the units of angular velocity given?

Angular velocity is given as the amount of central angle through which the rider travels over a given amount of time.

The central angle swept out by a rider traveling once around the wheel is 360°, or 2Π radians.

If one trip around the wheel takes 20 minutes, then the angular velocity of a rider is:

_2Πrad_ = _Π_ radians per minute

20 min       10                           

§3.5

What is the formula for linear velocity?

P is a point on a circle 

r is the radius of a circle

s is the distance P moves

(along the circumference of a circle)

t is the amount of time in which P moves

then linear velocity, v, of P

is given by the formula:

v = _s_

      t

§3.5

What is the formula for angular velocity?

P is a point moving with uniform circular motion

r is the radius of the circle

θ is the central angle that forms as r sweeps out

t is the amount of time

Then angular velocity, ω (omega), of P is given by the formula:

ω=_θ_

    t

§3.5

What is the relationship between the two velocities (angular and linear)?

If we start with the equation that relates

arc length, s, and central angle measure, rθ:

s=rθ

Divide both sides by time, t:

_s_=_rθ_

t      t  

_s_=r  θ_

t        t

v=rω

§3.5

Which formula relates linear velocity, v,

and angular velocity, ω?

Linear velocity is the product of the radius and the angular velocity:

v=rω

§3.5

How are conversion factors used to convert

inches per minute

into

feet per minute?

from 848 inches per minute, to feet per minute:

848 _in = 848_in   1_ft_= 848ft/min≈70.7ft/min

    min    1min   * 12in     12                         

§3.5

What is a gear train?

§4.1

Periodic Functions

A function y=f(x)

is said to be periodic

with period p

if p is the smallest positive number

such that f(x+p)=f(x)

for all x in the domain of f

EXAMPLE

1. because sin(x+2π)=sinx:
1. the funcion y=sinx is periodic with period 2π.
2. Likewise, because tan(x+π)=tanx:
1. the funcion y=tanx is periodic with period π.

§4.1,4.2

Amplitude

The amplitude A of a curve

is half the abosolute value

of the difference between

the largest value of y, denoted by M,

and the smallest value of y, denoted by m.

A=(1/2) l M-m l

EXAMPLE

1. the function y=5sin[πx+(π/4)]:
1. A=(1/2) l 5-(-5) l
2. A=(1/2) (10)
3. A=5

§4.1

Basic Graphs

sine and cosine

 The graphs of y=sinx and y=cosx

• period 2π.

• amplitude is 1.

• Sine curve

passes through 0 on the y-axis[image]

• Cosine curve

passes through 1 on the y-axis[image]

§4.1

Basic Graphs

Cosecant and Secant

 The graphs of y=cscx and y=secx

• period 2π.

• No amplitude:

There is no largest or smallest value of y

• Graphed as reciprocals of sine and cosine.
• the secant function:[image]
• The cosecant function:[image]

§4.1

Basic Graphs

Cotangent and Tangent

 The graphs of y=tanx and y=cotx

• period π.
• No amplitude
• Tangent curve passes through the origin:

[image]

• Cotangent is undefined when x is 0:

[image]

§4.1

Even and Odd Functions

• An even funcion is a function for which f(-x)=f(x) for all x in the domain of f
• An odd function is a function for which f(-x)=-f(x) for all x in the domain of f
• Cosine is an even funcion, and sine is an odd funcion.
• That is, cos(-θ)=cosθ cosine is an even function
• and sin(-θ)=-sinθ Sine is an odd function.
• The graph of an even function is symmetric about the y-axis, and the graph of an odd function is symmetric about the origin.

EXAMPLE

1. Tanθ is an odd function:
1. tan(-θ)={[sin(-θ)]/[cos(-θ)]}
2. tan(-θ)=[(-sinθ)/(cosθ)]
3. tan(-θ)=-(sinθ/cosθ)
4. tan(-θ)=-tanθ

§4.3

Phase Shift

The phase shift for a sine or cosine curve is the distance the curve has moved right or left from the curve y=sinx or y=cosx. 

EXAMPLE

We usually think of the graph of y=sinx as starting at the origin.  If we graph another sine curve that starts at (π/4), then we say this curve has a phase shift of (π/4).

[image]

§4.2-3

Graphing Sine and Cosine Curves

The graphs of:
y=A sin (Bx + C)
y=A cos (Bx + C),
where B>0,

• Will have the following characteristics:

1. Amplitude= l A l
2. Period= (2π/B)
3. Phase Shift= -(C/B)

• To graph one of these curves:
• construct a frame for one cycle.

• To mark the x-axis:
• Find where the cycle begins and ends by solving 0≤Bx+C≤2π for x.
• The left endpoint will be the phase shift.
• Find the period and divide it by 4 and use this value to mark off four equal increments on the x-axis.

• To mark the y-axis:
• Use the amplitude.

• Sketch in one complete cycle of the curve in question:
• Keeping in mind that if A is negative, the graph must be reflected about the x-axis.

y = sin x and overlay with y = 4 sin (2x +1)

[image]

§4.3

Vertical Translations

• Adding a constant, k, to a trigonometric function:
• translates the graph vertically up or down. 

EXAMPLE

• The graph of y=k+Asin(Bx+C)
• will have the same shape, if indicated, as y=Asin(Bx+C)
• Shape, meaning: amplitude, period, phase shift, and reflection
• y=k+Asin(Bx+C)
• will be translated k units
• vertically from the graph of y=Asin(Bx+C)

[image]

§4.6

Graphing by Addition of y-coordinates

• To graph equations of the form y=y₁+y₂,
• where Y₁ and Y₂ are alegbraic or trigonometric functions of x:

1. we graph y₁ and y₂ seperately on the same coordinate system.
2. Then add the two graphs to obtain the graph of y.

§4.7

Inverse Trigonometric Function

y = sin^-1 x

or

y=arcsin x

meaning:

x=sin y

and

-(π/2)≤y≤(π/2)

In words:

y is the angle between -(π/2) and (π/2), inclusive, whose sine is x.

§4.7

Inverse Trigonometric Function

y=cos^-1 x

or

y=arccos x

Meaning:

x=cos  y

and

0≤y≤π

In words:

y is the angle between 0 and π, inclusive, whose cosine is x.

§4.7

Inverse Trigonometric Function

y=tan^-1 x

or

y=arctan x

Meaning:

x=tan y

and

-(π/2)<y<(π/2)

In words:

y is the angle between -(π/2) and (π/2) whose tangent is x.

§4.7

Inverse Trigonometric Functions

EXAMPLES

• sin^-1(1/2)
• acrcos [-(√3)/2]

• sin^-1(1/2)
• The angle between -(π/2) and (π/2)
• whose sine is (1/2)
• is (π/6)
• sin^-1(1/2)=(π/6)

• acrcos [-(√3)/2]
• The angle between 0 and π
• with a cosine of [-(√3)/2]
• is (5π/6)
• acrcos [-(√3)/2]=(5π/6)

§3.3

REVIEWED:

What is a function?

A function is a rule

that pairs each element of the domain

with exactly one element from a range. 

§3.3

Why must the calculator always be in radian mode when evaluating circular functions?

The circular functions are functions of real numbers,

so they must always be evaluated on a calculator

in radian mode.

§3.3

When are the tangent and secant functins undefined?

The tangent and secant functions

are both undefined when X=0, because:

• tant=y/x
• sect=1/x

Which will occur at the points (0,1), and (0,-1).

or when t=π/2+πk, where k is any integer.

§3.3

When are the cotangent and cosecant functions undefined?

The cotangent and cosecant functions will be undefined when y=0, because:

• cott=x/y
• csct=1/y

Which will occur at the points (1,0) and (-1,0)

or when t=πk, where k is any integer.

§3.3

What is the domain for the circular function cost?

The domain of the circular function for cost is

All real numbers

(-∞,∞)

§3.3

What is the domain for the circular function tant?

The domain of the circular function for tant is

All real numbers except

t=π/2+kπ

for any integer k

§3.3

What is the domain for the circular function sect?

The domain of the circular function for sect is

All real numbers except

t=π/2+kπ

for any integer k

§3.3

What is the domain for the circular function cott?

The domain of the circular function for cott is

All real numbers except

t=kπ

for any integer k

§3.3

What is the domain for the circular function csct?

The domain of the circular function for csct is

All real numbers except

t=kπ

for any integer k

§3.3

What is the range of the circular function cost?

[-1,1]

§3.3

What is the range of the circular function tant?

All real numbers

(-∞,∞)

§3.3

What is the range of the circular function cott?

All real numbers

(-∞,∞)

§3.3

What is the range of the circular function sect?

(-∞,-1]υ[1,∞)

§3.3

What is the range of the circular function csct?

(-∞,-1]υ[1,∞)

§3.5

What is the relationship between

the radius of a circle

and the linear velocity of a point on the circle?

If there is a point (M) half way between

the center of a circle

and the point (P) on the outter edge of a circle.

Then M relates to P:

1. angular velocity is equal.
2. linear velocity is the product of the radius and the angular velocity.  Since there are two different radius lengths, there are two different linear velocities:

• multiply the linear velocity of P by (1/2) to obtain the linear velocity of M.

§5.1

Proving Identities

• An identity in trigonometry is a statement that two expressions are equal for all replacements of the variable for which each expression is defined.

• To prove a trigonometric identity, we use trigonometric substidutions and algebraic manipulations to either:

1. Transform the right side into the left side,
2. or transform the left side into the right side.

• Remember to work on each side seperately.  We do not want to use properties from algebra that involve both sides of the identity-like the addition property of equality.

• To prove tanx + cosx = sinx (secx + cotx):
• We can multiply through by sinx on the right side and then change to sines and cosines.

1. sinx (secx + cotx) = sinx secx + sinx cotx
2. =sinx * (1/cosx) + sinx * (cosx/sinx)
3. =(sinx/cosx) + cosx
4. =tanx + cosx

§5.2

Sum and Difference Formulas

sin(A + B)

§5.2

Sum and Difference Formulas

sin(A - B)

§5.2

Sum and Difference Formulas

cos(A + B)

§5.2

Sum and Difference Formulas

cos(A - B)

§5.2

Sum and Difference Formulas

tan(A + B)

§5.2

Sum and Difference Formulas

tan(A - B)

§5.2

Sum and Difference Formulas

Example

Find the exact value for cos75°

• To find the exact value for cos75°:
• write 75° as (45° + 30°)
• Apply the formula for cos(A + B)

1. cos75° = cos(45° + 30°)
2. = cos45° cos30° - sin45° sin30°
3. ={[(√2)/2][(√2)/2]}-{[(√2)/2](1/2)}
4. [(√6) - (√2)] / 4

§5.3

Double-Angle Formulas

sin2A

§5.3

Double-Angle Formulas

cos2A (3 forms)

1. First form:  cos2A = cos²A - sin²A
2. Second form:  cos2A = 2cos²A - 1
3. Third form:  cos2A = 1 - 2sin²A

§5.3

Double-Angle Formulas

tan2A

§5.3

Double-Angle Formulas

EXAMPLE

sinA = (3/5) and A is in QII

PROVE:  cos2A = (1 - 2sin²A)

cos2A = (1 - 2sin²A):

1. = 1 - 2sin²A
2. = 1 - 2(3/5)²
3. =(7/25)

§5.4

Half-Angle Formulas

sin(A/2)

§5.4

Half-Angle Formulas

cos(A/2)

cos(A/2) = ± √[(1 + cosA) / 2]