Chapter 4.5 and 4.6

Okay so for 4.5 we learned about the basic since and cosine curves.The cosine and since graphs look like this:

So the x-coordinates on this graph represents the angle measure in radians on a unit circle, and the y-coordinates represents the points at which the angle equals the x-coordinate (if that makes sense). The cosine and sine functions are pretty much identical, except for the fact that the intercepts are different because they have 0s at different points on the graph. So as you can see from the graph, the sines and the cosines all have the same max and min, the same period length. The only difference is that the the y value of sine = 0 when x = 0, while the y value of cosine = 1 when x=0. This is because the sine represents the y value of the point on a unit circle, and

the cosine represents the x value of the point.

A period in the a graph is 2 pis long for a sine and cosine.

The equation for these graphs are in this format:

y = d+a sin(bx-c) or y = d+a cos(bx-c)

d moves the graph up and down

a vertically stretches and compresses a graph, meaning that it will increase or decrease the maximum and minimum, but leave the x intercepts alone.

b horizontally stretches and compresses a graph, meaning that it will stretch out the x intercepts or compress them together

c moves the graph left and right, but unlike d, c moves right when it is subtracting from x, and moves left when adding to x.

The next section, 4.6, was about the tangent function, cotangent, cosecant, and secant functions

The tangent function is the sine function over the cosine function, which looks like this:

This means that whenever the x makes the value of the cosine = 0, the answer for the tangent function will be undefined, meaning there will be an asymptote wherever cosine = 0. This also means that wherever sine = 0, the tangent value will also be 0, so that's where the graph will intercept the x axis.

Unlike cosine and sine, the period is only 1 pi.

The cotangent function is the cosine over sine, so for this when the sine value = 0, there is a asymptote and when the cosine value = 0, there is an intercept. They look like this:

Cosecant functions are parabolas that open up at the max and open down at the min of a sine function. Their asymptotes are when x= n(pi) and they look like this:

The equation for these are: csc x = 1/sinx

Finally, the last one is the secant function. This one is like the c

osecant, because it looks just like it except that it opens up and opens down at the max and min of a cosine function. Their asymptotes are when x = (pi) / 2 + n (pi)

The graph for this looks like this:

Inverse Trigonometric Functions

For a function to have an inverse, it must pass the Horizontal Line Test. As shown below, y=sinx does not pass the test.




The graph is one-to-one only from peak to trough.

Because of this, in order to have an inverse function, we must restrict the domain to the interval:

.

The range would then be [-1,1]



So the restricted domain of , y=sinx has an inverse called the inverse sine function...

The graph would look something like this...



***The output of an inverse trigonometric function is an angle. In this case the output of the inverse sine would be in either the first or the fourth quadrant.***

The cosine function is decreasing on the interval as shown below




As with the sine function, we must also restrict the domain of the cosine function in order to have an inverse cosine function.

The restricted domain would be: [-1,1]
The range would be:

This graph would look like this....

***Again, remember the output of an inverse trigonometric function is an angle. So for this case the output of the inverse cosine function would be in the first or second quadrant.***





Similar to the inverse sine and cosine functions, you can define an inverse tangent function also by restricting the domain.

We restrict the domain of the inverse tangent function to all real numbers.
The range being .

The graph would like like this....




***Don't forget the output of an inverse trigonometric function is an angle. In this case the output of the inverse tangent lies in the first or fourth quadrant.***

Section 4.4 Trigonometric Functions of Any Angle

Blogpost 3rd Hour Pre Calc.

Today we learned about Reference Angles and Inverse Trigonometric Functions

REFERENCE ANGLES

• The values of the trigonometric functions of angles greater than 90° (or less than 0° ) can be determined from their values at corresponding acute angles called REFERENCE ANGLES.

• Definition: Letθ be an angle in standard position. Its REFERENCE ANGLE is the acute angleθ'formed by the TERMINAL SIDE ofθ and the HORIZONTAL AXIS.

• In short terms, Reference Angles are always Acute and every angle has one. The reference angle is the angle formed between the horizontal X-axis and the terminal side of the angle that intersects the origin.

In order to find reference

angles you need to determine which quadrant you want the reference angle to be in.

Different Reference Angles exist in all of the 4 quadrants. The

way to find out the reference angle in the designated quadrant is in the following

formulas.

In quadrant 1

:

1=

In quadrant 2:

1=-

In quadrant 3:

1=-

In quadrant 4:

1=2-

Today we also learned about identity inverses:

For example......

If you were assigned to find the angle measure algebraically when given the equation sin

= 1 you wouldn't be able to find solve and find the correct angle measure.

Today we learned that by taking the sine or cosine inverse of whatever the equation equals, we can find the 2 separate corresponding angle measures.

For example sin

= 1

In order to determine

you would plug in sin-1(1) and it would give you 90 degrees or /2 radians.

Almost always, with the exception of 1,-1, and 0, there is a corresponding value in the circle where the sin-1 value is the same. One example is when sin = 1/2, there will be two angle measures; one in the first quadrant and one in the second quadrant where both sin values are positive.

Trigonometric Identities

4.3 -- Right Triangle Trigonometry

The Six Trigonometric Functions

There are three different lengths of a triangle. One is the side opposite theta, the second is the side adjacent theta, and the last is the hypotenuse (opposite of the right angle). The lengths of these sides can be used to determine the Six Trigonometric Functions which include:

Sine- Opposite/Hypotenuse

Cosine- Adjacent/Hypotenuse

Tangent- Opposite/Adjacent

Cosecant- 1/Sine or Hypotenuse/Opposite

Secant- 1/Cosine or Hypotenuse/Adjacent

Cotangent- 1/Tangent or Adjacent/Opposite

Here is an example of a right triangle, where "A" is theta.

Example:

Hyp-5

Opp-3

Adj-4

Given these numbers for the side lengths and the definitions above, we can find out any of these Six Trigonometric Functions like so:

Sin= 3/5

Cos= 4/5

Tan= 3/4

Csc= 5/3

Sec= 5/4

Cot= 4/3

The Basics of Trigonometry - 4.1

So what exactly is trigonometry?

Trigonometry is the measurement of triangles.
Well triangles have angles right? What are those you ask?

Angles are two rays with a common endpoint, which is called a vertex (plural--vertices). Every angle has an intial side, the starting position of the ray, and a terminal side, the position after rotation. Such an angle is in standard postion (see diagram to right). A positive angle is one that comes as a result of counterclockwise rotation. A Negative angle is one that results from clockwise rotation. Some of the angles that were discussed in class today are:

Acute - less than 90

Right - 90

Obtuse - greater than 90 but less than 180

Sraight - 180

Reflex - greater than 180

Complementary - (not to be confused with complimentary) - two angles whose sum is equal to 90

Supplementary - two angles whose sum is equal to 180

Since math always has to make things harder than they really need to be (just kidding Mr. Wilhelm...but not really), we no longer use degrees to measure angles; we use radians. But wait, what is a radian? One radian is the point at which the intercepted arc is equal to the radius of the circle. It can be represented by the simple equation s=r, where "s" is the intercepted arc and "r" is the radius.

Next, we talked about Coterminal Angles. Coterminal Angles are angles that have the same initial side and terminal side.

When changing from degrees to radians:

Multiply the degree measure by /180

When changing from radians to degrees:

Multiply the radian measure by 180/

Chapter 2 Section 4: Complex Numbers

Today we learned about Complex Numbers in Section Four.

Before we did that, though, we reviewed the realm of Real numbers. Real numbers include both Rational numbers (e.g. 4, -7, -7/3, 0), and Irrational numbers (e.g. ,  , 5+)

Next, Mr.Wilhelm started the section on complex numbers.  He described them as having a real part and an imaginary part.

Standard Form of a complex number:  a+bi (where a and b are both real numbers and i represents the imaginary).

We were reminded of the value of an imaginary unit:

                           i=

Addition and Subtraction of Complex Numbers:

When adding complex numbers, one treats the imaginary unit like any other variable.

                        (2+4i)+(6-5i)= 8-i

the same is true when subtracting..

                        (2+4i)-(6-5i)= -4+9i

Multiplication of Complex Numbers:

When multiplying complex numbers, the important thing to remember is that i^2= -1

                        (2+4i)(6-5i)=

                      12 +14i-20i^2=         you multiply -20 by -1 and get +20

                        12+14i+20=

                            32+4i

(always simplify and make sure your answer is in standard form)

Dividing Complex Numbers:

When dividing Complex numbers, the key thing to remember is to multiply the numerator and denominator of your problem by the conjugate of the denominator.

                         now multiply by the conjugate

next, distribute and get... 

next, simplify the i^2s to -1s...

next, combine like terms and get...

finally, convert into standard form...

After learning about operations of Complex Numbers, Mr.Wilhelm talked a little bit about how even if a graph (e.g. a parabola) does not touch the x-axis, it still has imaginary zeros.

Then, we talked about taking i to higher powers.

 i^1= i

i^2=-1

i^3= -i  (because root -1 times root -1 times root -1 is equal to negative root -1 which is equal to -i)

i^4= 1 (because root -1 times root -1 times root -1 is equal to negative root -1, and when you multiply that by another root -1 you get -i^2 which is equivalent to 1)

we also talked about how this applies to other higher powers in a cycle of fours but I don't really understand that yet so I probably shouldn't confuse all of you..

We finished off the day talking about imaginary numbers as zeros, converting the zeros into polynomials and vice versa.

*If something like 5+3i is a zero, then its conjugate (5-3i) is also a zero

Example (creating a polynomial out of zeros and imaginary zeros):

Zeros: 2, 5+3i, and therefore 5-3i

the long way is...

f(x)= (x-2)(x-(5+3i))(x-(5-3i))

=(x-2)(x-5-3i)(x-5+3i)

=(x-2)[x^2-5x+3ix-5x+25-15i-3ix+15i-9i^2]

=(x-2)[x^2-10x+25-9i^2]

=(x-2)[x^2-10x+25+9]

=x^3-10x^2+34x-2x^2+20x-68

=x^3-12x^2+54x-68

the shorter way is...

(starting from the second step of the last example)

f(x)=(x-2)(x-5-3i)(x-5+3i)

(x-5-3i) represents (a-b) and (x-5+3i) represents (a+b), and when you take (a-b)(a+b) you get a^2-b^2

so in actuality you have f(x)=(x-2)(x-5)^2 -(3i)^2 (because x-5 represents a and 3i represents b)

this simplifies to

f(x)=(x-2)(x^2-10x+34)

=x^3-10x^2+34x-2x^2+20x-68

=x^3-12x+54x-68