Everything Terrific About Trig by Madison Thompson

Everything Terrific About Trig by Madison Thompson https://padlet.com/madisonethompson12/34we9u4l1jx1 Made with good vibes en-us 2020-02-13 23:48:53 UTC 2020-02-16 23:38:51 UTC hello@padlet.com Converting from Degrees to Radians madisonethompson12 https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/445360401 This is important for when you need to reference radians from degrees, or give an answer in degrees and radians that you found in degrees. It's found by the two ways below:

1) Multiplying by pi/180
2)
a) Dividing by 180
b) Converting to fraction
c) Multiplying the numerator by pi]]> 2020-02-13 23:57:00 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/445360401 Converting From Radians to Degrees madisonethompson12 https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/445361036 As noted above, this is useful for comparing radians to degrees, converting to degrees to perform operations, or when an answer demands both forms. You can do this two ways:

1) Multiply by 180/pi
2)
a) divide by pi
b) multiply by 180]]> 2020-02-13 23:59:40 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/445361036 Coterminal Angles madisonethompson12 https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/445374005 For the math people: coterminal angles are angles that share the same initial side and terminal side.

For the non-math people: angles that are intervals of 360 or 2pi apart.

So pretty much, coterminal angles are angles that share the same initial and terminal side, and they can be found by adding or subtracting in intervals of 360 degrees or 2pi.

Examples:
1) Find two coterminal angles of 180 degrees. One positive, and one negative.

Positive:
180+360=540

Negative:
180-360=-180

540 and -180 degrees.

2) Find two coterminal angles of pi. One positive, and one negative.

Positive:
pi+2pi=3pi

Negative:
pi-2pi=-pi

3pi and -pi radians.

You must always use operations of 360 degrees or 2pi.

More Examples:
https://www.chino.k12.ca.us/site/handlers/filedownload.ashx?moduleinstanceid=5575&dataid=2608&FileName=NOTES%20COTERMINAL%20ANGLES.pdf

https://www.youtube.com/watch?v=6BRtPfofXog

https://www.mathwarehouse.com/coterminal-angle/how-to-calculate-coterminal-angles.php]]> 2020-02-14 00:41:40 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/445374005 Reference Angles madisonethompson12 https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/445378289 Math words: Reference angles are the angles opposite the terminal side of a designated angle that are relative to the x-axis.

More basic: Reference angles are the angles that are found by using the value relative to the x-axis of the angle not included in the angle itself that is of the lowest value.

Most basic: the angle of the smallest value relative to the terminal side of the angle that is in contact with the x-axis.

Examples:
If an angle had a value of 190 degrees, what is the value of the reference angle?
10 degrees

If an angle had a value of 70 degrees, what is the reference angle?
70 degrees

If an angle had a value of 290 degrees, what is the reference angle?
70 degrees

More Info:
https://www.mathwarehouse.com/trigonometry/reference-angle/finding-reference-angle.php

https://www.youtube.com/watch?v=wT4xMAssvDk

https://www.youtube.com/watch?v=Flx0m5CIrpE]]> 2020-02-14 00:55:21 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/445378289 DMS Form madisonethompson12 https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/445392479 DMS is degree-minute-second form. DMS form is used for looking at degrees as a form of time, or as a form of degrees.

Converting to DMS:
First, keep the degrees.

Then, take the decimal and multiply it by 60. That is the minutes.

Then, take the remaining decimal and divide it by 60. This is the seconds.

One hash (') represents minutes.
Two hashes (") represent seconds.

Example:
45.8565 degrees.

45 degrees

.8565/60=51.39

45 degrees and 51 minutes

.39*60=23.4

45 degrees 51'23"

More Info:
https://www.youtube.com/watch?v=Flx0m5CIrpE

https://www.youtube.com/watch?v=Flx0m5CIrpE

Convert from DMS form to Degree Form:
Take the degrees out

Divide the minutes by 60

Divide the seconds by 3600

Add the two values together, they become the decimal of the degree

Example:
45.65'55"

45 degrees

.65/60=.01083

.55/3600=.00015278

.01083+.00015278=.0109861

45.012 degrees

More Info:
https://www.youtube.com/watch?v=WAxQtYe6_xg

https://www.youtube.com/watch?v=EGEzvlaUEUw]]> 2020-02-14 01:40:04 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/445392479 Arc Length https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/445402951 Arc length is a portion of the circumference of the circle that is found from a given angle.

Pretty much, it's the portion of the circumference that is found from multiplying the value of the radius by the value of the angle, in radians.

Here's more info information about what it is.
https://en.wikipedia.org/wiki/Arc_length

This the the formula to remember: S=r(-)(theta)

Example:
If a circle has a radius of five inches and an angle of pi/4 radians.

(5)(pi/4)

3.92 inches

More Info:
https://www.khanacademy.org/math/geometry/hs-geo-circles/hs-geo-arc-length-deg/v/length-of-an-arc-that-subtends-a-central-angle

https://www.khanacademy.org/math/geometry/hs-geo-circles/hs-geo-arc-length-deg/v/length-of-an-arc-that-subtends-a-central-angle]]> 2020-02-14 02:15:13 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/445402951 Sector Area https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/445406880 The area of a portion of a circle, given the radius and the angle (in radians).

The formula for this is A=1/2(-)r²A Quick Video Lesson:
https://www.youtube.com/watch?v=cAOVS2DTU0UExample:
If a circle has an radius of 3 feet and a central angle of 85 degrees, find the area.

(1/2)(85)(3x3)

382.5 feet squared

More Info:
https://revisionmaths.com/advanced-level-maths-revision/pure-maths/trigonometry/radians

https://www.khanacademy.org/math/geometry/hs-geo-circles/hs-geo-sectors/e/areas_of_circles_and_sectors]]> 2020-02-14 02:28:22 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/445406880 Basic Trig Ratios https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/445410840 In a right triangle, we can use SOHCAHTOA to find values relevant to theta, or the angle of interest.

Sine is equal to
Opposite over
Hypotenuse

Cosine is equal to
Adjacent over
Hypotenuse

Tangent is equal to
Opposite over
Adjacent

These trig ratios are the basic ones, and they can be learned more about here:
https://revisionmaths.com/gcse-maths-revision/trigonometry/sin-cos-and-tan

Or here: https://www.mathsisfun.com/sine-cosine-tangent.html

SOHCAHTOA is used for finding the side value that corresponds to the ratio paired to theta. For example, the side that is the tangent of theta is a ratio equal the length of the opposite side over the length of the adjacent side. By finding the value of tan and then supplementing x for the value you want to find, you can algebraically solve for the side length. Here is a worked example of sin, cos, and tan.
https://www.youtube.com/watch?v=XFh_JC7OSrg

In addition, you can also use the trig ratios to find the angle values. This is done by using the inverse trig functions, or the functions that are trig to the negative first power. These functions are called ArcSin, ArcCos, and ArcTan. They can be used by finding the ratio of side values, then inputing that ratio into the Arc function of the ratio you want to find. The result will be the angle value. For example, if you had an ArcSin function with the ratio of 29/35, you would input it into sin of negative one and get the value of 55.95, the angle you needed to find. Here is another example:
https://www.youtube.com/watch?v=y3LajBcnb7s

And another:
https://www.youtube.com/watch?v=3TQM6j31xUc

These are the reciprocal functions.
These functions can also be written as reciprocals of secant, cosecant, and cotangent, the functions shown below.

Cosecant (csc) is the reciprocal of sin

Secant (sec) is the reciprocal of cos

Cotangent (cot) is the reciprocal of tan

More Info:
https://www.google.com/search?q=secant%2C+cosecant+and+contangent&oq=secant%2C+cosecant+and+contangent&aqs=chrome..69i57j0l7.8861j1j4&sourceid=chrome&ie=UTF-8

https://www.khanacademy.org/math/trigonometry/trigonometry-right-triangles
In trig identities these functions are reciprocals of each other, as seen below.]]> 2020-02-14 02:42:02 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/445410840 45-45-90 Triangles madisonethompson12 https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/445664818 45-45-90 triangles are unique in that they also have set values for their lengths. As can be seen below, their side values are: s, s, and s times the square of two.

These ratios can be useful for quickly finding the values of triangles, without using the Pythagorean theorem. This concept can be further elaborated on by the video below:
https://www.brightstorm.com/math/trigonometry/pythagorean-theorem/45-45-90-triangles/

Example:
The leg of a right triangle with a 45 degree angle has a value of 5. What is the value of the remaining three sides.

First, this question forces you to infer that the triangle is 45-45-90. However, once inferred this information makes solving the triangle much easier.

s=5, so the other leg must be equal to 5 as well.

The hypotenuse is equal to five times the square root of two, or five radical three.

So, the side values are 5, 5, and 5 root 2.

To understand why 45-45-90 triangles are the way they are, look here:
https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-special-right-triangles/v/45-45-90-triangle-side-ratios

In addition, special right triangles are relevant to the unit circle. The unit circle, linked here: https://www.youtube.com/watch?v=YMe8NCuC95I is used to find trig ratios.

On the 45-45-90 triangle, it is related to the coordinates of square root of two over two. Because the values are divided by each other to equal one (s), they are multiplied by the square root of two to find the hypotenuse (the value that connects the center of the circle to the opposite end of the hypotenuse).

For more info on this proof, reference below:
https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-special-right-triangles/v/45-45-90-triangles

https://study.com/academy/lesson/45-45-90-triangle-theorem-rules-formula.html

]]> 2020-02-14 16:23:55 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/445664818 30-60-90 Triangles madisonethompson12 https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/445680094 30-60-90 is the last special right triangle. 30-60-90 triangles have side lengths (shown on the image below) with values of s, (the shortest leg), s times root 3 (the longer leg) and 2s, (the hypotenuse).

For a general overview of these triangles, look here: https://study.com/academy/lesson/30-60-90-triangle-theorem-properties-formula.html

Example:
If a right triangle with an angle value of 30 degrees has a shorter leg with a value of 4, then what are the values of all sides?

Again, this problem makes us infer that the triangle is 30-60-90. However, once we know that it's a 30-60-90, we can find the other sides easily.

Side s is given, and is 4.

The hypotenuse is equal to 2s, or 2(4). This is equal to 8.

Finally, the value of the longer leg is equal to root 3 times s. So, the longest leg is equal to four root two.

The side values are 4, 8, and 4 times the square root of 3.

30-60-90 triangles are also linked to the unit circle, through both the 30 degree and the 60 degree points.

As is shown on the unit circle, the longest leg is in correspondence with the square root of three over two, and one half. When these are divided by each other to find values, it corresponds directly to root three, which is the value we use to find the length of the leg. The shortest side is s, the base.

This is the proof:
https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-special-right-triangles/v/30-60-90-triangle-side-ratios-proof

And this is more information:
https://blog.prepscholar.com/30-60-90-triangle-ratio-formula

https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-special-right-triangles/v/30-60-90-triangle-example-problem]]> 2020-02-14 16:50:12 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/445680094 Why Law of Sines and Cosines? madisonethompson12 https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446012507 We need these laws for when triangles aren't right triangles, and we cannot use the normal trig ratios. Law of sines is beneficial for when we have only side values or only one angle, while law of cosine is better for when we have an abundance of angles. These correspond with our triangle side ratios from math two, as linked below:
https://tutors.com/math-tutors/geometry-help/triangle-congruence-theorems-sss-sas-asa

https://mathbitsnotebook.com/Geometry/CongruentTriangles/CTtriangleMethods.html

These laws can also be used together when convenient, to find angles and sides. However, it's normally quicker and easier to use just one per problem. This is an example with both:
https://www.youtube.com/watch?v=NJ4_RUQI014]]> 2020-02-16 02:30:10 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446012507 Law of Sines madisonethompson12 https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446013529 2020-02-16 02:37:11 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446013529 Law of Sines madisonethompson12 https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446013626 As is seen in the image above, the law of sines states that the length of side a divided by the sine of the measure of angle A is equal to side b divided the sine of angle B is equal to side c divided by the measure of angle C.

To simplify that, each side on the triangle divided by the sin of its corresponding angle is equal to the same formula from every other side.

Here's some more explanation:
https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-law-of-sines/v/law-of-sines

https://www.mathsisfun.com/algebra/trig-sine-law.html

Really, you just plug in all the given values and then solve algebraically to find the unknowns. Here is an example.

Example:
If a triangle has side values of 50, 40, and 80, with one known angle of 35 degrees, corresponding to the side with value 40, what is the value of the angle opposite value 80?

First, you would divide the value of 40 by sin(35).

This is 69.74.

Then, you would divide by 80, giving you .8715.

Finally, you find the ArcSin of that value, as you are solving for the angle. This is equal to 60.66, the measure of the angle opposite the side of value 80.

More Info:
https://www.varsitytutors.com/hotmath/hotmath_help/topics/law-of-sines

https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-law-of-sines/v/law-of-sines-example

https://www.mathwarehouse.com/trigonometry/law-of-sines/formula-and-practice-problems.php]]> 2020-02-16 02:38:06 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446013626 Law of Sines and Cosines madisonethompson12 https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446017090 https://www.shelovesmath.com/trigonometry/law-sines-cosines/

https://www.cut-the-knot.org/pythagoras/cosine2.shtml

https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-solving-general-triangles/a/laws-of-sines-and-cosines-review

https://www2.clarku.edu/faculty/djoyce/trig/laws.html]]> 2020-02-16 03:05:34 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446017090 Law of Cosines madisonethompson12 https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446017430 2020-02-16 03:08:07 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446017430 Law of Cosines madisonethompson12 https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446017509 As can be seen in the image above, law of cosines is a law that states that the length of side a squared is equal to side b squared plus side c squared minus two times side b and c, times cosine of angle a.

The side that is set equal to the rest of the equation is the side with the corresponding angle that is being found the cosine of.

Law of cosines proof:
https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-law-of-cosines/v/law-of-cosines

https://www.mathopenref.com/lawofcosinesproof.html

This proof analyzes the law of cosines, and how it works when angles are split to create multiple right triangles.

Example:
You are given a triangle with the following side values a=x, b=10, c=12. You know the value of angle A to be 55 degrees. Find the value of side a.

You don't know the value of side a, but you are solving for it. Therefore, it becomes x squared.

x^2 = 10^2 + 12^2 - 2(10)(12)(cos(55))

x^2 = 100 +144 - 240(.57)

x^2 = 244 - 136.8

x^2 = 107.2

Now, you find the square root of both sides to isolate x.

x = 10.35

For more worked examples, links are here:
https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-law-of-cosines/v/law-of-cosines-example

https://sciencenotes.org/law-of-cosines-example-problem/

https://www.youtube.com/watch?v=VTNHGeyR0ow]]> 2020-02-16 03:08:48 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446017509 Graphing Sine Functions madisonethompson12 https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446075189 Graphing any trig function can be related to the unit circle. When you're graphing the parent sine function, the x-axis corresponds to the radians of the unit circle, while the y-axis corresponds to the values of the y-coordinate.

This video explains this concept in more depth:
https://www.youtube.com/watch?v=hUxBdhCRHK4

This an example of the table of values for the parent function of sine.

X _________Y
0 _________0
pi/2_______1
pi ________ 0
3pi/2_____-1
2pi _______ 0

This is one cycle, then it repeats each time you go around the circle. This is an example of the sine parent function.

It shows two periods, and as you can see each value will correspond to the unit circle.
X to radians.
Y to the y-coordinate.]]> 2020-02-16 13:19:58 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446075189 Graphing Cosine Functions madisonethompson12 https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446076799 Cosine graphs are the same as sine graphs in that the x value is derived from the radians. However, they derive their y-values from the x-coordinate. This means that they will be a little different, starting with the fact that their values start at 1.

Before continuing, watch and read these resources to make sure that you understand:
https://www.mathopenref.com/triggraphcosine.html

https://www.youtube.com/watch?v=yCfMLKcwJqQ

https://www.youtube.com/watch?v=SdHwokUU8xI

This is the table for the values of the parent function of cosine:

X _________Y
0 _________1
pi/2_______0
pi ________-1
3pi/2_____ 0
2pi _______1

As you can see, the x-values remain the same. The only thing that changes is the y-value, as the cosine values pull from the x-coordinate, and the sine values pull from the y-coordinate. In general, sine is equal to y, x is equal to cosine, and tangent is equal to sine over cosine, or y/x.

The graph below shows the cosine parent function on two periods (two times around the unit circle). When graphing cosine, remember:
x is equal to radians.
y is equal to x-coordinates.]]> 2020-02-16 13:35:30 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446076799 Graphing Tangent Functions madisonethompson12 https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446078295 Tangent functions are the most difficult to understand. Tan functions do use the radians for their x-values as well, however they also have asymtopes at every 1/2 pi. Therefore, the period of tangent is only pi, as opposed to sine and cosine, which have periods of two pi.

Asymptotes- points on a graph that a function will approach, but never reach.

Periods- the amount of time it takes a cycle to repeat. In sine and cosine, they go around the entire unit circle with different values, so they repeat every two pi (360 degrees). In tan functions, they repeat every pi due to their vertical asymtopes. This gives them a period of pi.

Why does tangent have asymtopes at pi/2 and 3pi/2?

As you can recall from previous slides, the tangent function is equal to y/x. Each coordinate has varying negatives and positives of 0 and 1. So, for the of pi/2 and 3pi/2, the tangent is 1/0 and -1/0, respectively. This is undefined, so tangent is undefined at those radian values.

This causes it to repeat every pi, instead of every two pi.

Another important thing about tangent graphs is their shape. Tangent graphs look similarly to cubic function. The extend until reaching their amplitude, which for the parent function is one unit up, and one unit over for the side that is on the positive side of the x-axis (if at the origin) and one unit down and one unit over for the negative side. The amplitude value, which is technically not present in tangent functions, is met when the function is halfway to the asymptote. Once reaching this value, they extend vertically indefinitely.

These resources explain tangent graphs:
https://www.youtube.com/watch?v=hUrjUDDzECA

https://www.youtube.com/watch?v=n0cVo54OfXI

https://www.purplemath.com/modules/triggrph2.htm

This is the table of values for tangent functions:

X _________Y
0 ________ positive, negative infinity
pi/2_______ asymptote
pi _______ positive, negative infinity
3pi/2_____ asymptote
2pi ______ positive, negative infinity

This the parent function of trig functions graphed.
Note the following:
x is derived from radians
y is derived from y/x coordinates

]]> 2020-02-16 13:47:18 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446078295 Translations of Trig Functions madisonethompson12 https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446082030 Definitions:

Amplitude- the highest point the graph reaches from the midline.

Period- the time it takes for the cycle to go through everything, before it will repeat itself.

Phase Shift- how far left or right a trig function moves.

Vertical Shift- how far up or down a trig function moves.

Formula:
For F(t) = A f(Bt – C) + D, where f(t) is one of the basic trig functions, we have:

the amplitude is |A|

the period is 2 pi/ B for sine and cosine, but pi/B for tangent

the phase shift is C/B

the vertical shift is D

So all of this is great, but how do we apply it?

First, the amplitude will be what y-value each parent function will reach instead of one. So, if the amplitude has a value of five, the functions would translate as follows:

Sine:
X _________Y X _________Y
0 _________0 0 _________0
pi/2_______1 pi/2_______5
pi ________ 0 pi ________ 0
3pi/2_____-1 3pi/2_____-5
2pi _______ 0 2pi _______ 0

Cosine:
X _________Y X _________Y
0 _________1 0 _________5
pi/2_______0 pi/2______ 0
pi ________ -1 pi _______-5
3pi/2_____ 0 3pi/2_____ 0
2pi _______ 1 2pi _______ 5

Tangent will reach the points (1,5) and (-1, 5). It does not have an amplitude.

Why doesn't tan have an amplitude?
https://www.varsitytutors.com/hotmath/hotmath_help/topics/graphing-tangent-function

Amplitudes of sin and cosine:
https://www.varsitytutors.com/hotmath/hotmath_help/topics/amplitude-and-period-of-sine-and-cosine-functions

https://www.youtube.com/watch?v=rGD_56tU-NM

Period is explained above, however to retouch it's just the amount of points it takes for a cycle to repeat, or the number of points that it takes for the cycle to go around the unit circle.
sine and cosine have a period of 2pi.
tan has a period of pi.

More links on periods of trig functions:
https://www.youtube.com/watch?v=SBqnRja4CW4

https://www.youtube.com/watch?v=yZLeN_9jjGg

Phase Shift
The phase shift is the amount of space that the graph moves to the right or to the left. The tables below show how a phase shift of pi/2 would change the graph.

Sine:
X _________Y X _________Y
0__________0 pi/2______ 0
pi/2 ______ 1 pi ________ 1
pi_________ 0 3pi/2_____ 0
3pi/2 ____ -1 2pi _______-1
2pi _______ 0 5pi/4_____ 0
Cosine:
X _________Y X _________Y
0_________ 1 pi/2______1
pi/2 ______0 pi _______ 0
pi________-1 3pi/2_____ -1
3pi/2_____ 0 2pi _______ 0
2pi_______ 1 5pi/4_____ 1
Tan:
X _________Y X _________Y
0__+ or - infinity 0___asymptote
pi/2_asymptote pi/2_+ or - infinity
pi _+ or - infinity pi _asymptote
3pi/2_asymptote 3pi/2+ or - infinity
2pi __+ or - infinity 2pi _asymptote

Phase Shift notes (cummulative):
http://www.onemathematicalcat.org/Math/Precalculus_obj/amplitudePeriodPhaseShift.htm

https://www.youtube.com/watch?v=crF8IHXpQu8

Vertical Shift
This shows if the midline, normally the x-axis is moved up or down for the function. It's important to remember that the amplitude is based on the midline, not the x-axis.

Vertical shift notes:
https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/graph-functions-using-vertical-and-horizontal-shifts/

https://study.com/academy/lesson/vertical-shift-definition-examples.html

This is an image showing vertical shifts:

]]> 2020-02-16 14:22:32 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446082030 Review of Composition of Functions madisonethompson12 https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446090139 Composition of functions is when instead of x, the variable of one function is another function. To review, look at this example:

f(x) = x + 2
g(x) = 8x + 9

What is f(g(x))?

Well, you must plug g(x) into each value of x for f(x). In this example, f(x) has only one spot for a variable, so g(x) is only inserted once.

f(g(x)) = (8x+9) + 2

8x + 9 + 2

f(g(x)) = 8x +11

More review:
https://mathbitsnotebook.com/Algebra2/Functions/FNComposition.html

https://www.purplemath.com/modules/fcncomp3.htm]]> 2020-02-16 15:20:02 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446090139 Composition of Trig Functions madisonethompson12 https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446090891 In trig functions, things can get more complicated. Because all trig functions are really ratios that can be related back to each other, trig function compositions will often result in a value on the unit circle.

To do composition of trig, you first have to understand basic trig. This example will show you how to derive coordinates from the trig and the value given.

Remember this for the rest of your math career:
x is equal to cosine
y is equal to sine
y/x is equal to tangent

Example:
what is cos(90 degrees)

So, cos(90 degrees) is equal to pi/s radians.

The x-value of pi/2 radians is equal to 0, and because cosine is equal to the x-value, cos(90 degrees) is equal to 0.

Example 2:
sin(90 degrees)

Radians of this is equal to pi/2.

The y-coordinate of pi/2 is equal to 1.

sin(90 degrees) is equal to 1

Example 3:
tan(90 degrees)

Radians of this is equal to pi/s.

Y/X for this coordinate is equal to 1/0, or undefined.

tan(90 degrees) is undefined

The concept shown above can be found by referencing the unit circle, and is shown in the following video:
https://www.youtube.com/watch?v=NO4H4YROdqk

Now that we've gone over this, it seems to get more complicated but really it's no different. When we want to solve equations such as sin(cos(300)) it's just like having two of the equations that we illustrated above. So, after we solve cos(300), which is equal to 1/2, we solve sin(1/2), using inverse sine to isolate points. The only points on the unit circle where sin is equal to 1/2 is pi/6.
These resources give more examples and information:
https://www.shelovesmath.com/trigonometry/trig-inverses/

https://www.ck12.org/trigonometry/composition-of-trig-functions-and-their-inverses/lesson/Composition-of-Trig-Functions-and-Their-Inverses-TRIG/

https://www.andrews.edu/~rwright/Precalculus-RLW/Text/04-09.html

Example 2:
cos(sin(3pi/4))

sin(3pi/4) is equal to the square root of two over two

cos(root 2/2)

cos(root 2/2) is equal to pi/4

cos(sin(3pi/4)) is pi/4

Inverse functions can also be used. So, you could have a problem asking you to find the inverse sine of cosine of 45 degrees. This is how you would solve that problem.

sin^-1(cosine(45))

cosine(45) is equal to root 2/2

sin^-1(root 2/2) is equal to 45 degrees, or root 2/2.

Inverse of any trig will cancel out the value of the the trig function it is. So, sin^-1(sin(90)) will equal 90, as the sines cancel out.

More on inverses:
https://www.youtube.com/watch?v=pWdGu9E5nCE

These functions can be solved from radians or degrees. They have definite correspondence to the unit circle. ]]> 2020-02-16 15:26:51 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446090891 Graphing the Reciprocal Functions madisonethompson12 https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446131149 Each reciprocal function does have its own way to be graphed, however sine and cosine follow the same system. Here's the steps:

1) Graph the function as it's reciprocal. So for cosecant, graph sine. For secant, graph cosine.
2) Draw asymptotes where the graph crosses the x-axis.
3) Draw parabolas that are reflections over the x-axis of the function you've already drawn. They should start at the minimum or maximums that are on the graph, and continue indefinitely with mind to the asymptotes.
4) Erase the function drawn in step one.

Cotangent is just tangent reversed. The end on the side approaching positive infinity on the x-axis approaches negative infinity on the y-axis, and vice versa.

More Notes:
https://www.youtube.com/watch?v=zpIMLztQrS0

https://www.youtube.com/watch?v=kpBNFv8HJyY]]> 2020-02-16 19:45:25 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446131149 Graphing Inverse Functions madisonethompson12 https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446134130 Graphing inverse trig functions means that restrictions have to be placed. Otherwise, the functions wouldn't be one-to-one. They're just like the normal functions, just only in set intervals.

This is how we set intervals. Shocker, it involves the unit circle:
https://www.youtube.com/watch?v=Gvo7ILkNssE

Stated restrictions:

ArcSin-- (-pi/2, pi/2)

ArcCos-- (0, pi)

ArcTan-- (-pi/2, pi/2)

Here's even more notes:
https://www.varsitytutors.com/hotmath/hotmath_help/topics/inverse-trigonometric-functions]]> 2020-02-16 20:04:27 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446134130 Composition of Trig Functions madisonethompson12 https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446137102 https://math.libretexts.org/Bookshelves/Precalculus/Book%3A_Precalculus_(OpenStax)/06%3A_Periodic_Functions/6.04%3A_Inverse_Trigonometric_Functions

https://www.youtube.com/watch?v=NnqhDbZlqNI

https://www.youtube.com/watch?v=6Q8LnlRnR1Q

https://www.youtube.com/watch?v=pNgkK_MR6jM]]> 2020-02-16 20:22:58 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446137102 Graphing Trig madisonethompson12 https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446137915 https://www.youtube.com/watch?v=vHYI93UV5Kg

https://www.nctm.org/Classroom-Resources/Illuminations/Interactives/Trigonometric-Graphing/

https://www.desmos.com/calculator/nlfjfdf3kp

https://www.khanacademy.org/math/trigonometry/trig-function-graphs/basic-graphs-of-sine-cosine-tangent/v/we-graphs-of-sine-and-cosine-functions?modal=1]]> 2020-02-16 20:27:14 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446137915 Trig Functions madisonethompson12 https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446138230 https://www.youtube.com/watch?v=F21S9Wpi0y8

https://www.youtube.com/watch?v=PUB0TaZ7bhA

https://www.youtube.com/watch?v=aRVWs1tDarI

https://study.com/academy/lesson/reciprocal-functions-definition-examples-graphs.html

https://www.youtube.com/watch?v=nVTtSE5nv7c]]> 2020-02-16 20:29:03 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446138230 Angles madisonethompson12 https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446140505 https://mathbitsnotebook.com/Geometry/Circles/CRArcLengthRadian.html

https://iitutor.com/arc-length-and-sector-area/

https://webpages.uidaho.edu/learn/math/lessons/lesson03/3_01.htm

https://www.youtube.com/watch?v=E-xFXpVo14o

https://www.youtube.com/watch?v=pt-anJytCJ0

https://www.youtube.com/watch?v=E-xFXpVo14o

https://www.varsitytutors.com/hotmath/hotmath_help/topics/coterminal-angles]]> 2020-02-16 20:41:53 UTC https://padlet.com/madisonethompson12/34we9u4l1jx1/wish/446140505