Graphing Inverse Trigonometric Functions

Graphing inverse Trigonometric functions is pretty much like taking a part of a trigonometric function that passes the horrizontal line test, and flipping it over the line y=x

this is what arctan looks like....

tanx                arctan

D:(-pi/2,pi/2)                  D:(-infin,infin)

R:(-infin,infin)                   R:(-pi,pi)

this is what arcsin looks like (just the red part)...

sinx                arcsinx

D:(-pi/2,pi/2)                  D:(-1,1)

R:(-1,1)                   R:(-pi/2,pi/2)

and this is what arccos looks like (just the black part);

cosx                arccosx

D:(0,pi)                  D:(-1,1)

R:(-1,1)                   R:(0,pi)

Remember, the output of an inverse trig function is an ANGLE

Graphing of Trigonometric Functions

Today in class we learned how to graph all the trigonometric functions.

y = sin x

Domain: all real numbers

Range: [-1, 1]

Period: 2π

y = cos x

Domain: all real numbers

Range: [-1, 1]

Period: 2π

y = tan x

Domain: all, x can't equal π/2 + n(π)

Range: (-∞, ∞)

Period: π

y = csc x = 1/ sin x

Domain: all x can't equal n(π)

Range: (-∞, -1] and [1, ∞)

Period: 2π

y = sec x = 1/ cos x

Domain: all x can't equal π/2 + n(π)

Range: (-∞, -1] and [1, ∞)

Period: 2π

y = cot x = 1/tan x

Domain: all x can't equal n(π)

Range: (-∞, ∞)

Period: π

4.4 Trigonometric Functions of Any Angle

Today in class we learned how to find the six trigonometric functions for any angle.
Defenitions of Trigonometric Functions of Any Angle
Let θ be an angle in standard position with (x,y) a point on the terminal side of θand r = √x^2+y^2 = 0.

sinθ = y/r                 cosθ = x/r
tanθ = y/x                cotθ= x/y
secθ = r/x                cscθ = r/y

To evluate these trigonometric functions, when given a point on the terminal side of θ, plug in the coordinates of that point to find r. Then plug in r and the afforementioned cordinates for the above defenitions and have fun.


We also discussed in class the signs of the trigonometric functions in each quadrant.

Quadrant I            Quadrant II            Quadrant III            Quadrant IV
sinθ: +                    sinθ: +                      sinθ: -                        sinθ: -
cosθ: +                   cosθ: -                     cosθ: -                       cosθ: +
tanθ: +                    tanθ: -                      tanθ: +                       tanθ: -

4.3 Right Triangle Trigonometry

Today we learned about right triangle trigonmetry and the six trigonometric functions. Consider a right traingle, one of whose acute angles is labeled theta. Relative to the angle theta, the three sides of the triangle are the hypotenuse, the opposite side, and the adjacent side.

Note that the functions in the second row above are the reciprocals of the corresponding functions in the first row.
Find the exact values of the six trigonmetric functions of theta.

sin theta= 4/5

cos theta=3/5

tan theta=4/3

csc theta=5/4

sec theta= 5/3

cot theta=3/4

Note that sin 30 =1/2=cos 60. This occurs because 30 degrees and 60 degrees are complementary angles, and, in general, it can be shown from the right triangle definitions that confunctions of complementary angles are equal. That is, if theta, is an acute angle, the following relationships are true.

sin(90 degrees-theta)= cos theta

tan(90 degrees-theta)=cot theta

sec(90 degrees-theta)=csc theta

cos(90 degrees-theta)=sin theta

cot(90 degrees-theta)=tan theta

csc(90 degrees-theta)=sec theta

4.1 Radian & Degree Measure

Trigonometry: Radian and Degree Measure

When the length of a segment of the circumference of a circle is equal to the radius, this is a measurement known as a radian. There are 2π radians in a circle.


There are 2π radians in one full revolution. 2π radians = 360º

To convert degrees to radians, you multiply the degree by π/180

For example, take 45º 45 • π/180 = 45π/180 = π/4

45º is the same measurement as π/4 radians


You can look at a circle as either 360º or 2π radians. Because it is all proportional, π radians is the same as 180º, π/2 radians is the same as 90º, π/4 is the same as 45º, etc.




Cosine is the x coordinate of the endpoint and sine is the Y coordinate.

When two measurements land on the same line, these measurements are called co-terminal. For example, π/4 radians is the same as 9π/4 radians, and because of this, they are co-terminal.

2.6 Rational Functions

Note: I could not upload pictures because of maintenance.

Today we learned about rational functions. This involved the format for a rational function which is,

f(x)=N(x)/D(x), where N(x) and D(x) are polynomials
 Ex: 2x^2+x-1/3x-1
 (Keep in mind D(x) cannot be to the degree of 0 otherwise it is just a normal polynomial)

Due to maintenance I could not insert graphs to show what this equation and other equations look like.

We also learned about the tendencies of these equations

ex. f(x)=2x/x-3

Here when x-> +/-infinity      f(x)->2
also when x->3       f(x)->+/-infinity

One of the major things we learned about in class were vertical and horizontal asymptotes which keep the points on the curve of the graph from going way too high, so that it would be almost impossible to measure.

Some rules of asymptotes that we learned were:
 The closer x gets to the vertical asymptote, the larger the y or f(x) value.

The larger x gets, the closer the y value is to the horizontal asymptote.

To find the vertical asymptotes, you take the denominator of the rational function and make it equal to zero. So to find the vertical asymptote of the equation above, we would do this:

x-3=0
x=3

Given this the vertical asymptote is 3.

To find the horizontal asymptote, there are three cases of how to find it.

Case One:  when a lower degree polynomial/higher degree polynomial, than the horizontal asymptote is 0.
ex. 2x/x^2-3
horizontal asymptote: y=0

Case Two: when to of the same degree polynomials are divided by each other, you take the coefficients of each of the first terms and divide those by each other. Once that is found, that is the horizontal asymptote.

ex. 2x/x-3
2/1=2 ----> horizontal asymptote: y=2

Case Three: when a higher degree polynomial/lower degree polynomial, there is no horizontal asymptote.

ex. 2x^2/x-3
No horizontal asymptote

2.4 complex numbers

In this chapter we solve complex equations by adding, subtracting, multiplying, and dividing.
we use the imaginary number i which is defined as the square root of -1.


If a and b are are real numbers, the number a + bi is a complex number. which is supposed to be written in standard form a+bi = c+di
iff a=c and b=d

If b is not = to zero, the number a+bi is an imaginary number

(4+4i) + (3-6i) = 7-2i addition of complex numbers

(4+4i) - (3-6i) = 1+10i subtraction of complex numbers

(4+4i)(3-6i) = 12 -12i -24i^2 i^2 =-1 -24(-1) =24 multiplication of complex numbers
= 36-12i

4+4i/3-6i multiply numerator and denominator by 3+6i to cancel out middle term
= -12+36i/9-36i^2

-12/45 + 36/45i
= -4/15+ 4/5 i (written in standard form) division of complex numbers


Expanding
zeros: 2, 3-4i, and 3+4i

f(x)= (x-2)(x-(3-4i))(x-(3+4i))
f(x)= (x-2)(x-3+4i)(x-3-4i)
(a + b) (a - b) = a^2 -b^2
f(x)= (x-2)(x-3)^2-(4i)^2

f(x)= x^3-8x^2+37x-50 (expanded)