**Trig Graphs and Identities**

Trig Graphs

The first thing about Trigonometric graphs is than any function that repeats itself in a fashion that can be measured is called periodic. Examples of these are sine and cosine. In math, a function is a set of inputs with a set of related outputs.

To measure a period, we must be familiar with the unit circle, shown here.

A full period, or trip around the unit circle then, is pi to 2pi.

These are the graphs for sine and cosine for one period, or one trip around the unit circle.

There are two more terms we need to know when discussing trig graphs. The first is amplitude. This is the measure of a graphs lowest point to its highest divided by two. I look at graphs like geographic figures. Measure the distance between the highest mountain and lowest valley and take half of it.

When you take the absolute value of this number and it is less than 1, it is compressed vertically.

The next term is phase shift. A phase shift is a horizontal translation of a trigonometric function.

The rules with which to remember these shifts is if c>0, the graph shifts to the left and if c<0, the shift is to the right given the formula shift = -c/b

The last shift is a vertical translation which can be done by adding a constant in front of a function. The is the same for the functions that we covered in the very last unit.

The maximum and minimum values center around one line called a midline. This line fluctuates depending on how the function is altered by vertical translations.

The frequency of a sine or cosine function refers to the number of times it repeats compared to the parent function's period. Frequency and period of have a reciprocal relationship with one another.

maximum

midline

minimum

period

amplitude=a

Intercepts

7,0

.75pi,0

Examples:

A) In this function, y = -2 sin x, the amplitude is 2 because of the 2 value in front of the function and the period is a normal 2pi because no horizontal translations have been made.

B) In this function, y = 3 sin 10x, the amplitude is 3 because of the corresponding A value and the period is 2pi/10 = pi/5.

This graph is a prime example of a veritcal shift as well as horizontal compression. The function for this graph is y = -3 + 5cos (pi x/12).

midline at -3 shows vertical translation

There's more!

We have more graphs! The trigonometric identity tanx=sinx/cosx can be turned into a graph. Using simpler terms, the function is y=tanx. The only problem with this graph is that when cosx is 0 we have an issue. You can't have a zero in the denominator of a fraction, and cosx is 0 for any multiple of pi/2. Because of this y = tan x has vertical asymptotes when x =pi/2(n), where n is an odd integer. The range of this function is all real numbers, eliminating amplitude as a factor. This creates a funky looking graph, which reminds one of waves.

asymptote

y=tanx

all real numbers

The cotangent function is just the inverse of the tangent function. Because of this the graph is reversed.

The next graph is cosecant

cscx=1/sinx

We have another case where we have issues if the denomenator is zero. When sinx is zero the graph has vertical asymptotes when x = npi.

The domain of y = csc x is all real numbers except x = pi n.

The range is all real numbers greater than or equal to 1 or less than or equal to -1. The period is the same as y = sin x, which is 2π.

Here is the graph:

asymptote

The secant graph is the inverse of the cosecant graph. Since secant is undefined when cos x = 0, the graph has vertical asymptotes when x =pi/2(n).

The domain is all real numbers except x = pi/2(n).

The range is all real numbers greater than or equal to 1 or less than or equal to -1. The period is the same as y = cos x, which is 2pi.

transformation

transformation

Lastly we have inverse trigonometric function that are called arcsin arctan and arccos.

sin-1 tan-1 cos-1

Luckily we have a graph to determine the range and domain of each of these functions.

Example:

A) find the value of arcsin -1/2

To solve this question we are basically trying to find the sign of what is -1/2

By using our unit circle we can find that the value of this is -pi/6 because sin -pi/6 = -1/2

change in period

change in period

**Trig Identities**

An identity is an equation which is true for every value of the variable. An example of this would be something like a(b)=b(a).

We can break down trigonometric identities into 5 sections, the first being Pythagorean identities.

1. Pythagorean Identities

Obviously the Pythagorean therom is a^2+b^2=c^2

By using this formula we're able to get identities from this.

sin^2theta+cos^2theta=1

tan^2theta+1=sec^2theta

cot^2theta+1=csc^2theta

2. Quotient Identities

tantheta=sintheta/costheta

cottheta=costheta/sintheta

3.

These identities lay the foundation for solving problems.

By substituting each identity into problems we're able to simplifiy and switch things around.

Example

cotx/cscx

1. cot x is equal to cos x/sin x

2. csc x is equal to 1/sin x

3. This turns into the equation (cos x/sin x)/(1/sin x)

4. This equals cos x

We can see here why its important that we know all of the identities or have a graph of them.

4. Next we have sum and difference identities. These identities help us calculate more sines and cosines of angles.

By breaking down angles that we know into smaller angles we can answer questions.

Example:

Evaluate Sin 75deg

1.We can break down the sin of 75 into smaller well know angles like 30 and 45

2. This will give us sin(30+45)

3. The rest we just plug into the equation.

4. sin(30 +/- 45)=sin30 cos45 +/- cos30 sin45

5. Plug these into your calculator or use your brain if you have them memorized!

6. 1/2(sqrt2/2)+sqrt3/2(sqrt2/2)

= (sqrt2+sqrt6)/4

5. Double angle identities

By using the double angle formula, we can rewrite functions and make them easier to solve.

Example:

Rewrite 4 – 8 sin^2 x

1. 4 – 8 sin2 x

2. = 4(1 – 2 sin2 x) = 4((cos2x + sin2x) – 2 sin2x)

3. = 4 (cos2x – sin2x)

4. = 4 cos 2x

I've included a video that helped me a lot when trying to understand graphs and how to draw them.

**Again just adding a video that really helped me when trying to figure out some things. This guys website is patrickjmt.com and it literally has videos on every thing you could think of involving math. Thank you so much Ms. Barbour, I hope you enjoy!**

The End