Graphs of Other Trigonometric Functions

- Cosecant graphs have NO x- intercepts

- The vertical asymptote is where sin = 0

- The period is

- Secant graphs have NO x- intercepts

- The vertical asymptote is where cos = 0

- The period is

- Where sin = 0, tan is the x- intercept

- Where cos = 0, tan is the vertical asymptote

- The Period is

- Where cos = 0, cot is x- intercept

- Where sin = 0, cot is the vertical asymptote

- The period is

Trigonometric Functions of Any Angle

In this section, the definitions of trigonometric functions are extended to cover any angle.

In trigonometry, we tend to think of an angle as created by a rotating ray. The beginning position of the ray is called the initial side. Usually, the initial side coincides with the positive x-axis; angles with such an initial side are said to be in standard position. The post-rotation position of the ray is called the terminal side. The measure of the angle is a number which describes the amount of rotation. If the rotation of the ray was clockwise, the angle measure is a negative number; if the rotation was counterclockwise, the angle measure is positive.


An angle θ in standard position creates a right triangle on the coordinate axes as follows:

1. Choose a point P0=(x0, y0) on the terminal side of the angle.
2. Draw a vertical line from P0 to the x-axis (note: this line is perpendicular to the x-axis).
3. The resulting right triangle with vertices (x0, y0), (x0, 0), (0,0) is called the reference triangle for the angle θ

The vertical side of this triangle has length y0; the horizontal side has length x0; the hypotenuse is the distance from P0 to the origin, or




Trigonometric Functions of any angle


















*Note: Notice that tan(θ) and sec(θ) are not defined when the terminal side of the angle is vertical; Also cot(θ) and csc(θ) are not defined with the terminal side of the angle is horizontal.





Evaluating Trigonometric Functions When Given a Point

To evaluate a trigonometric function just take the point given on the terminal side of the angle and plug it into the formula for the function you solving listed above in the table. You can think of r as the hypotenus of the reference triangle so

Sine and Cosine Graphs

Amplitude: is half the distance between the maximum and minimum on the graph. (height of one wave)

- Amplitude = |a|

Period: the length from one crest to the next (wavelength)

- 2π/b

Translations: y= a x sin (b (θ - c)) + d

• Y creates a vertical change

• A creates a vertical change. The larger the number, the longer the stretch becomes

• B creates a horizontal change. The larger the number, the more compressed the graph becomes.

• θ creates a horizontal change

• C creates a horizontal change. It shifts to the left if positive, and the right when negative.

• D causes the entire function to shift up or down Y creates a vertical change

Phase Shift: Sin(θ + 2π) = Cos

Sine Graph:

• Crosses through the origin
• Amplitude is 1
• Period is 2π

Cosine Graph:

• Starts at 1
• Amplitude is 1
• Period is 2π

· Identities: an equation that is always true.
o Reciprocal Identity:
§ Ex)\[sin\theta =1/csc\theta \]
· (applies to all six)
o Quotient Identity:
§ Ex)\[tan\theta =sin\theta /cos\theta \]
· (applies to all six)
o Even Odd Identities
§ Ex)\[sin(-\theta )=-sin\theta \]
· (applies to all six)
o Pythagorean Identities
§ Three different identities
· 1) \[sin^2\theta +cos^2\theta =1\]
· 2) \[tan^2\theta +1=sec^2\theta \]
· 3) \[cot^2\theta +1=csc^2\theta \]
· Solving Identities:
o Ex) \[(cot\theta +csc\theta)(tan\theta -sin\theta)=sec\theta -cos\theta \rightarrow (1-cot\theta)(sin\theta)+(csc\theta)(tan\theta-1)\rightarrow -(cos\theta /sin\theta )(sin\theta )+(1/sin\theta)(sin\theta /cos\theta )\rightarrow -cos\theta +1/cos\theta \rightarrow sec\theta =cos\theta =sec\theta -cos\theta \]

Ratios

sine x = (side opposite x)/hypotenuse

cosine x = (side adjacent x)/hypotenuse

tangent x = (side opposite x)/(side adjacent x)

In the figure below, sin A = a/c, cosine A = b/c, and tangent A = a/b.

Reciprocal Ratios

cotangent x = 1/tan x = (adjacent side)/(opposite side) 

secant x = 1/cos x = (hypotenuse)/(adjacent side) 

cosecant x = 1/sin x = (hypotenuse)/(opposite side)

Cofunctions

sin x = cos (90o - x) 

tan x = cot (90o - x) 

sec x = csc (90o - x) 

cos x = sin (90o - x) 

cot x = tan (90o - x) 

csc x = sec (90o - x) 

Example:

1. Problem: Find the function value of

cot 60o.

Solution: Use the cotangent's cofunction

identity to rewrite the problem.

tan (90o - 60o)

tan 30o

The tangent of 30o is

(SQRT(3))/3

Section 4.1 Radian and Degree Measure

Angles

Radian Measure

Definition of a Radian

One radian is the measure of a central angle θ that intercepts an arc s equal in length to the radius r of the circle.

Example 1: Finding Coterminal Angles

Find a positive and a negative angle coterminal with a  angle.

A  angle and a  angle are coterminal with a  angle.

Section 2.7 Graphing rational functions

Graph f(x)=x/x^2-x-2

set the numerator equal to zero in order to find the x-intercepts

x-intercept = 0

plug in a zero to find the y-intercepts

y-intercept= 0

factor the denominator to find the vertical asymptotes

(be sure to write them as an equation of a line)

(x-2)(x+1)

x=2 x=-1

The horizontal asymptote is y=0 because the numerator degree is less than the denominator degree

Graph the function

(the graph will end up looking like the graph on the right)

Rules for horizontal asymptotes:

1. just use the leading term because the other terms are insignificant

2. If the degrees are the same for the leading term, just divide the numerator by the denominator

3. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote

4. If the numerator degree is less than the denominator degree, the horizontal asymptote is y=0

5. A line can cross the horizontal asymptote in the middle of the graph

6. written as y=

Vertical asymptotes are where the denominator is equal to zero

-you can never cross the vertical asymptote

-written as x=

Be sure to watch out for holes and zeros of multiplicity

F(x)= x (x-3)/ (x^2-x-2)(x-3)

(x-3) is in the numerator and denominator so you cross them out leaving a hole in the graph at x=3

When there is a zero of multiplicity of two, the graph will look exactly the same as the graph on the other side of the asymptote.

F(x)= x/(x-3)

x-intercept= 0

y-intercept=0

vertical asymptote is x=3

horizontal asymptote is y=1

2.6 Rational Functions and Asymptotes

Rational Functions can be written in the form:

Where N(x) and D(x) are polynomials and D(x) is not the zero polynomial.

Graphs of rational functions look like this...

Domain of a Rational Function: The domain of rational functions of x includes all real numbers except x-values, that will make the denominator zero.

(Eg. the function below has a domain of (-∞,3) U (3,∞))

Vertical and Horizontal Asymptotes:

Vertical asymptotes:

are found by setting the denominator equal to zero and solving D(x)=0.

(Eg. the function below has a vertical asymptote at x=3)

Horizontal Asymptotes:

There are three different scenarios for finding the horizontal asymptote. However, the first consideration that should be taken is that only the leading term on both the numerator and denominator when finding horizontal asymptotes. This is because asymptotes really only have an effect at the ends of the graph and due to the magnitude of the numbers at the end, the numbers following the leading term are insignificant.

Scenarios:

1. Leading terms are to an equal degree. When this occurs the answer is the constant in the leading term of the numerator over the constant of the leading term of the denominator.

(Eg. In the function below ignore everything after the leading term, the horizontal asymptote is y=3/4)

2. The leading term of the numerator is higher than the leading term of the denominator. In this x keeps growing and the function never levels off. There is no horizontal asymptote in this situation.

(Eg. In the function below there is no horizontal asymptote)

3. The leading term of the denominator is higher than the leading term of the numerator. In this x approaches zero and gets very tiny. The horizontal asymptote is y=0

(Eg. In the function below the horizontal asymptote is y=0)

Intercepts:

X-Intercepts:

To find the x intercepts of a rational function you must set y equal to zero, and therefore N(x)=0.

(Eg. the x-int for this function=0)

Y-Intercepts:

To find the y intercepts of a rational function you must set x equal to zero.

(Eg. the x-int for the function is 0.)
Things to Consider:

• Asymptotes are lines and therefore should be written in proper notation, such as x=a and y=b


• It is possible for a rational function to have multiple x intercepts and asymptotes


• It is possible for a rational function to lack both vertical and horizontal asymptotes


• It is possible for a rational function to lack both x and y intercepts

Complex Numbers Section 2.4

complex number- number consisting of a real part and an imaginary part.

complex numbers can be graphed as a pair of numbers forming a vector.

i is the imaginary unit defined as €i€€€ being equal to the square root of negative 1.

complex numbers are the broadest range of numbers consisting of imaginary numbers and all types of real numbers.

2.5 The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra:

If f(x) is a polynomial of degree n, where n > 0, f has a least one zero in the complex number system.

Linear Factorization Theorem:

If f(x) is a polynomial of degree n where n > 0, f has precisely n linear factors

Example1: Real Zeros of Polynomial Functions

• The first-degree polynomial f(x)= x-2 has exactly one zero: x=2
• Counting multiplicity, the second-degree polynomial function has exactly two zeros: x=3 and x=3

Example 2: Real and Complex Zeros of Polynomial Functions

• In some third degree polynomial functions there will be exactly three zeros, but the graph will only show the real zeros as intercepts.

Example 3: Finding the Zeros of a Polynomial Function

• To find the zeros, first find possible rational zeros then find the real zeros by graphing the equation, and finally use synthetic division to determine algebraically that the zeros are on the graph.

Complex Zeros Occur in Conjugate Pairs

Let f(x) be a polynomial function that has real coefficients. If a + bi, where b can't equal 0, is a zero of the function, the conjugate a-bi is also a zero of the function.

**this result is only true if the polynomial function has real coefficients.

Example 4: Finding a Polynomial with Given Zeros

• find a fourth-degree polynomial function with real coefficients, that has -1, -1, 3i as zeros
• from the Linear Factorization Theorem it can be written as
• or simply, let a=1 to get

Factors of a Polynomial:

Every polynomial of degree n > 0 with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros.

**a quadratic with no real zeros is irreducible over the reals

for example:

**this quadratic is irreducible over the rationals, but reducible over the reals:

Example 5: Factoring a Polynomial

a. as the product of factors that are irreducible over the rationals

• first factor the polynomial into the product of two quadratic polynomials

b. as the product of linear factors and quadratic factors that are irreducible over the reals

• By factoring over the reals, you have:

Example 6: Finding the Zeros of a Polynomial Function

given that 1-3i is a zero of f

• because complex zeros occur in conjugate pairs, you know that 1-3i is also a zero of f.
• both x-(1+3i) and x-(1-3i) are factors of f(x). Multiplying these two factors produces [x-(1+3i)][x-(1-3i)]=[(x-1)-3i][(x-1)+3i]

• then you would use long division to divide into f(x) and then get
• therefore, you have
• and conclude that the zeros of f are 1+3i, 1-3i, 3 and -2