Hi everyone, and welcome to MathSux! In this post, we are going to go over the Central Angles Theorems of circles. We’ll go over the theorems associated with central angles and then solve a quick example. Make sure to test your understanding of central angles and arcs with the practice questions at the end of this post. And, if you want more, don’t forget to check out the video below, happy calculating!

Central Angles and Arcs:

Central angles and arcs form when two radii are drawn from the center point of a circle. When these two radii come together they form a central angle. A central angle is equal to the length of the arc. When it comes to measuring the central angle, the central angle is always equal to arc length and vice versa:

Central Angles = Arc Length

Central Angle Theorems:

There are a two central angle theorems to know, check them out below!

Central Angle Theorem #1:

Central Angle Theorem #2:

Let’s look at how to apply these rules with an Example:

Let’s do this one step at a time.

Practice Questions:

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

Happy December everyone! With crazy 2020 coming to an end, I thought I would share some TikTok math video compilations of Algebra, Geometry, Algebra 2/Trig, and Statistics for a quick review of all our videos posted throughout the year. Enjoy these TikTok math video compilations and happy calculating! 🙂

Want to make math suck just a little bit less? Subscribe and follow us for FREE fun colorful math videos and lessons every week! 🙂

Within algebra, you will find arithmetic sequences, combining like terms, box and whisker plots, geometric sequences, solving radical equations, completing the square, 4 ways to factor quadratic equations, piecewise functions and more!

Geometry:

Within Geometry, you will find, how to construct an equilateral triangle, a median of a trapezoid, area of a sector, how to find perpendicular and parallel lines through a given point, SOH CAH TOA right triangle trigonometry, reflections, and more!

Algebra 2/Trig.

Within Algebra 2/Trig., you will find, how to expand a cubed binomial, how to divide polynomials, how to solve log equations, imaginary numbers, synthetic division, unit circle basics, how to graph y=sin(x), and more!

Statistics:

Within statistics, you will find, box and whisker plots, how to find the variance, and, the probability of flipping a coin 2 times!

For full length video, don’t forget to check out our free math video index page! Thanks for stopping by! 🙂

Hi everyone and welcome to MathSux! This post is going to help you pass Algebra 2/Trig. In this post, we are going to apply our knowledge of the unit circle and trigonometry and apply it to graphing trig functions y=sin(x), y=cos(x), and y=tan(x). We have been using trig functions in basic trigonometry, but today, we will see what each function (sin, cos, and tan) look like graphed on a coordinate plane.

If you are already familiar with the basics of graphing trig functions and want to learn how to transform them, including knowing how to identify the amplitude, period frequency, vertical shift, and horizontal phase shift, then please check out this post here on transforming trigonometric functions.

If you have any questions, don’t hesitate to comment or check out the video below. Thanks for stopping by and happy calculating! 🙂

How do we get coordinate points for graphing Trigonometric Functions?

For deriving our trigonometric function graphs [y=sin(x), y=cos(x), and y=tan(x)] we are going to write out our handy dandy Unit Circle. By looking at our unit circle and remembering that coordinate points are in (cos(x), sin(x)) form and that tanx=(sin(x))/(cos(x)) we will be able to derive each and every trig graph!

*Note below is the unit circle we are going to reference to find each value, for an in-depth explanation of the unit circle, check out this link here.

How to Graph y=sin(x)?

While Graphing Sine, you may notice that the sine function curves creates what looks like an “S” shape. As they say, “S” is fpr Sine, this is the easiest way to remember what the sine function looks like! Check it out below.

Now that we know what the sine function looks like once it’s graphed, let’s see why it looks that way in the first place by deriving each coordinate point with the help of the unit circle.

Step 1: We are going to derive each degree value for sin by looking at the unit circle. These will be our coordinates for graphing y=sin(x). *For a review on how to get these values, check out the link here explaining the unit circle.

Step 2: Now we need to convert all the x-values from degrees to radians. Fear not because this can be done easily with a simple formula!

To convert degrees à radians, just use the formula below:

Step 3: Now that we have our coordinate points and converted degrees to radians, we can draw out our function y=sin(x) on the coordinate plane!

Now we will follow the same process for graphing y=cos(x) and y=tan(x).

How to Graph y=cos(x)?

While Graphing Cosine, you may notice that the cosine function creates what looks like a “V” shape. I always like to think “V” is for victory, but of course, in this case, it is for cosine, that is the easiest way to remember what this function looks like! Check it out below.

Now that we know what the cosine function looks like once graphed, let’s see why it looks this way in the first place by deriving each coordinate point with the help of the unit circle.

Step 1: We are going to derive each degree value for cos by looking at the unit circle. These will be our coordinates for graphing y=cos(x). *For a review on how to get these values, check out the link here explaining the unit circle.

Step 2: Now we need to convert all the x-values from degrees to radians.

Step 3: Now that we have our coordinate points and converted degrees to radians, we can draw out our function y=cos(x) on the coordinate plane!

What do you think would happen if we were to vertical shift our entire function up by one unit? Or move our entire function to the left by 90 degrees with a horizontal phase shift? Check out what happens here!

How to Graph y=tan(x)?

While Graphing Tangent, you may notice that the tangent function looks totally different than sine or cosine. Check it out below.

Step 1: We are going to derive each degree value for the tangent function by looking at the unit circle. In order to derive values for tan(x), we need to remember that tan(x)=sin(x)/cos(x). Once found, these will be our coordinates for graphing y=tan(x). *For a review on how to get these values, check out the link here explaining the unit circle.

Step 2: Now we need to convert all the x-values from degrees to radians.

Step 3: Now that we have our coordinate points and converted degrees to radians, we can draw out our function y=tan(x) on the coordinate plane!

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above for an in-depth explanation. Happy calculating! 🙂

Ready for the next lesson? Now that you know the basics of graphing the sine, cosine, and tangent functions, learn how to transform each trig function on a coordinate plane by identifying the amplitude, period frequency, vertical shift, and horizontal phase shift of each function, by checking out this post here on transforming trigonometric functions.

r=Common Ratio (Number Multiplied/Divided to each Term in Sequence)

n= Term Number in Sequence

Hi everyone and welcome to Mathsux! In this post, we’re going to go over geometric sequences. We’ll see what geometric sequences are, break down their formula a_{n}=a_{1}r^{(n-1)}, and solve two different types of examples. As always if you want more questions, check out the video below and the practice problems at the end of this post. Happy calculating! 🙂

What are Geometric Sequences?

Geometric sequences are a sequence of numbers that form a pattern when the same number is either multiplied or divided to each subsequent term.

Example:

Notice we are multiplying 2 to each term in the sequence above. If the pattern were to continue, the next term of the sequence above would be 64. This is a geometric sequence!

In this sequence it’s easy to see what the next term is, but what if we wanted to know the 15^{th} term? That’s where the Geometric Sequence formula comes in!

Geometric Sequence Formula:

Now that we broke down our geometric sequence formula, let’s try to answer our original question below:

->First, let’s write out the formula:

-> Now let’s fill in our formula and solve with the given values.

Let’s look at another example where, the common ratio is a bit different, and we are dividing the same number from each subsequent term:

-> First let’s identify the common ratio between each number in the sequence. Notice each term in the sequence is being divided by 2 (or multiplied by 1/2 ).

-> Now let’s write out our formula:

-> Next let’s fill in our formula and solve with the given values.

Practice Questions:

Find the 12^{th} term given the following sequence: 1250, 625, 312.5, 156.25, 78.125, ….

Find the 17^{th} term given the following sequence: 3, 9, 27, 81, 243,…..

Find the 10^{th} term given the following sequence: 5000, 1250, 312.5, 78.125 …..

Shirley has $100 that she deposits in the bank. She continues to deposit twice the amount of money every month. How much money will she deposit in the twelfth month at the end of the year?

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

*Also, if you want to check out arithmetic sequences click this link here! And if you want to learn about even more sequences, check out the link here!

Greetings math friends! In today’s post we’re going to go over some unit circle basics. We will find the value of trigonometric functions by using the unit circle and our knowledge of special triangles. For even more practice questions and detailed info., don’t forget to check out the video and examples at the end of this post. Keep learning and happy calculating! 🙂

What is the Unit Circle?

The Unit Circle is a circle where each point is 1 unit away from the origin (0,0). We use it as a reference to help us find the value of trigonometric functions.

Notice the following things about the unit circle above:

Degrees follow a counter-clockwise pattern from 0 to 360 degrees.

Values of cosine are represented by x-coordinates.

Values of sine are represented by y-coordinates.

Using the unit circle we can find the degree and radian value of trigonometric functions (SOH CAH TOA). Check out the example below!

What’s the big deal with Quadrants?

Within a coordinate plane there are 4 quadrants numbered I, II, III, and IV used throughout all of mathematics. Within these quadrants there are different trigonometric functions that are positive to each unique quadrant. This will be important when solving questions with reference angles later in this post. Check out which trig functions are positive in each quadrant below:

Now let’s look at some examples on how to find trigonometric functions using our circle!

Negative Degree Values:

The unit circle also allows us to find negative degree values which run clockwise, check it out below!

Knowing that negative degrees run clockwise, we can now find the value of trigonometric functions with negative degree values.

How to find trig ratios with 30º, 45º and 60º ?

Instead of memorizing much, much more of the unit circle, there’s a trick to memorizing two simple special triangles for answering these types of questions. The 45º 45º 90º special triangle and the 30º 60º 90º special triangle. (Why does this work? These special triangles can also be derived and found on the unit circle).

Using the above triangles and some basic trigonometry in conjugation with the unit circle, we can find so many more angles, take a look at the example below:

Since we need to find the value of tan(45º) , we will use the 45º, 45º, 90º special triangle.

For our last question, we are going to need to combine our knowledge of unit circles and special triangles:

-> In order to do this, we must first look at where our angle falls on the unit circle. Notice that the angle 135º is encompassed by the pink lines and falls in quadrant 2.

-> Since our angle falls in the second quadrant where only the trig function sin is positive. Since we are finding an angle with the function cosine, we know the solution will be negative.

-> Now we need to find something called a reference angle. Which is what those θ, 180°-θ, θ-180°, 360°-θ and symbols represent towards the center of the unit circle. Using these symbols will help us find the value of cos(135º).

Because the angle we are trying to find,135º , falls in the second quadrant, that means we are going to use the reference angle that falls in that quadrant 180º-θ theta, using the angle we are given as θ.

-> Now we can re-write and solve our trig equations using our newly found reference angle, 45º.

Now we are going to use our 45º 45º 90º special triangle and SOH CAH TOA to evaluate our trig function. For a review on how to use SOH CAH TOA, check out this link here.

When you’re ready, try the problems on your own below!

Practice Questions:

Solve the following trig functions using a unit circle and your knowledge of special triangles:

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above for even more examples. Happy calculating! 🙂

d=Common Difference (Number Added/Subtracted to each Term in Sequence)

Hi everyone and welcome to Mathsux! In this post, we’re going to go over arithmetic sequences. We’ll see what arithmetic sequences are, break down the formula a_{n}=a_{1}+(n-1)d, and solve two different types of examples. As always if you want more questions, check out the video below and the practice problems at the end of this post. Happy calculating! 🙂

What are Arithmetic Sequences?

Arithmetic sequences are a sequence of numbers that form a pattern when the same number is either added or subtracted to each subsequent term.

Example:

Notice we are adding 2 to each term in the sequence above. If the pattern were to continue, the next term of the sequence above would be 12. This is an arithmetic sequence!

In the above sequence it’s easy to see what the next term is, but what if we wanted to know the 123^{rd} term? That’s where the Arithmetic Sequence Formula comes in!

Arithmetic Sequence Formula:

Now that we know the arithmetic sequence formula, let’s try to answer our original question below:

-> First, let’s write the arithmetic sequence formula:

-> Fill in our formula and solve with the given values.

Now let’s look at another example where we subtract the same number from each term in the sequence, making the common difference negative.

-> First let’s identify the common difference between each number in the sequence. Notice each term in the sequence is being subtracted by 3.

-> Now let’s write out our formula:

-> Next let’s fill in our formula and solve with the given values.

Practice Questions:

Find the 123^{rd} term given the following sequence: 8, 12, 16, 20, 24, ….

Find the 117^{th} term given the following sequence: 2, 2.5, 3, 3.5, …..

Find the 52^{nd} term given the following sequence: 302, 300, 298, …..

A software engineer charges $100 for the first hour of consulting and $50 for each additional hour. How much would 500 hours of consultation cost?

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

Greetings math folks! For anyone familiar with trigonometry and SOH CAH TOA trigonometric ratio you should know that there is something special about right triangles. We are about to learn more about right triangles, as there are two distinct types of special right triangles in this world that we need to know, this includes the 45 45 90 triangle and the 30 60 90 triangle. In this post, we are going to go over the 45 45 90 special right triangle! If you are looking for the other very famous special triangle, (30 60 90), check out this post here.

With the help of this special triangle, we are going to see how to find the missing sides of a right triangle when given only one of its lengths (and the angles of the right triangle given are 45 45 90). For even more examples, check out the video and practice questions below and at the end of this post. Happy calculating! 🙂

45 45 90 Right Triangle Ratio:

Looking above at our 45 45 90 special triangle, notice it contains one right angle and 2 equal angles of 45 degrees. Based on these angle proportions, we are able to infer information about each sides length, thats where our ratio comes in!

45 45 90 Triangle – Why is it special? Where did come from?

45 45 90 special right triangles are “special” because they are a type of Isosceles Right Triangle, meaning they have two equal sides (marked in blue below). If we know that the isosceles triangle has two equal lengths, we can find the value of the length of the hypotenuse by using the Pythagorean Theorem based on the other two equal sides. Check out how we derive the 45 45 90 ratio below!

Notice, we started with the Pythagorean Theorem, then filled in our variables based on each given sides length. Next, we combined like terms and then took the square root of each side of the equation. Lastly, we found the value of hypotenuse, c, based on the other two legs, which is equal to the length of a times radical 2.

Now that we know the length of the hypotenuse in terms of each sides length a, we can re-label our triangle. Since we found the length of the hypotenuse in relation to the two equal legs, notice that this creates a ratio that applies to each and every triangle out there!

How do I use this ratio?

Ok, great we have derived the 45 45 90 ratio, but what do I do with this thing and how do I use it?

Knowing the above ratio, allows us to find the length of any missing side of a 45 45 90 special triangle (when given the value of one of its sides).

Let’s now try some 45 45 90 right triangle examples with missing sides below:

Example #1:

Step 1: First, let’s look at our ratio and compare it to our given right triangle.

Step 2: Notice we are given the value of the bottom leg, a=8. Knowing this we can fill in each length of our right triangle based on the ratio of a 45 45 90 special triangle shown below:

Now let’s look at an example where we are given the length of the hypotenuse and need to find the values of the other two missing sides of a 45 45 90 right triangle.

Example #2:

Step 1: First, let’s look at our ratio and compare it to our given right triangle.

Step 2: In this case, we need to do a little math to find the value of a, based on the Pythagorean Theorem. See how we use the Pythagorean Theorem step by step below to find the value of missing sides represented by a.

Now try mastering the art of the 45 45 90 special triangle on your own with the practice problems below!

Practice Questions:

Find the value of each missing side length of each 45 45 90 right triangle.

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

Hey there math friends! In this post we will go over how and when to use synthetic division to factor polynomials! So far, in algebra we have gotten used to factoring polynomials with variables raised to the second power, but this post explores how to factor polynomials with variables raised to the third degree and beyond!

If you have any questions don’t hesitate to comment or check out the video below. Also, don’t forget to master your skills with the practice questions at the end of this post. Happy calculating! 🙂

What is Synthetic Division?

Synthetic Division is a shortcut that allows us to easily divide polynomials as opposed to using the long division method. We can only use synthetic division when we divide a polynomial by a binomial in the form of (x-c), where c is a constant number.

Example #1:

*Notice we can use synthetic division in this case because we are dividing by (x+4) which follows our parameters (x-c), where c is equal to 4.

Example #2: Factoring Polynomials

Let’s take a look at the following example and use synthetic division to factor the given polynomial:

Check!

The great thing about these questions is that we can always check our work! If we wanted to check our answer, we could simply distribute 2(x+1)(x+3)(x-2) and get our original polynomial, f(x)=2x^{3}+4x^{2}-10x-12.

Try the practice problems on your own below!

Looking to brush up on how to divide polynomials the long way using long division? Check out this post here!

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

Greetings math friends! In this post, we arere going to go over the formulas for Variance and Standard Deviation. We will take this step by step by explaining the significance of the variance and standard deviation formulas in relation to a set of data. Get your calculators ready because this step by step although not hard, will take some serious number crunching! Also, don’t forget to check out the video on standard deviation and variance below to see how to check your work using a calculator. Happy calculating! 🙂

If you’re looking for related formulas, Mean Absolute Deviation (MAD) and Expected Value, scroll to the bottom of this post! And if you’re interested we’ll also touch upon the difference between population variance and sample variance later in this post.

What is the Variance?

The variance represents the spread of data or the distance each data point is from the mean. When we have multiple observations in our data, we want to know how far each unit of data is from the mean. Are all the data points close together or spread far apart? What is the probability distribution? This is what the variance will help tell us!

Don’t freak out but here’s the formula for variance, notated as using the greek letter, sigma squared, σ^{2}:

where…

x_{i}= Value of Data Point

μ= mean

n=Total Number of Data Points]

(x_{i}-μ)=Distance each data point is to the mean

In plain English, this translates to:

Let’s try an example to find the standard deviation and variance of the data set below.

Step 1: First, let’s find the mean, μ.

Step 2: Now that we have the mean, we are going to do each part of our formula one step at a time in the table below.

Notice we subtract each test score from the mean, μ=78. Then we square the result of each subtracted test score to get the squared deviation of each data value, then finally sum all the squared results together.

Step 3: Now that we summed all of our squared deviations, to get 730, we can fill this in as our numerator in the variance formula. We also know our denominator is equal to 5 because that is the total number of test scores in our data set.

What is Standard Deviation?

Standard deviation is a unit of measurement that is unique to each data set and is used to measure the spread of data. The standard deviation formula happens to be very similar to the variance formula!

Below is the formula for standard deviation, notated as sigma, the greek letter, σ:

Since this is the same exact formula as variance with a square root, all we need to do is take the square root of the variance to find standard deviation:

Sample VS. Population

What is the difference between a sample vs. a population?

A population in statistics refers to an entire data set that at times can be humanly incapable of reaching.

For example: If we wanted to know the average income of everyone who lives in New York State, it would be almost impossible to reach every working person and ask them how much they make for a living.

To make up for the impossibility of data collection, we usually only survey a sample of the entire population to get income levels of let’s say 10,000 people across New York State, a much more reasonable in terms of data collecting!

And taking this sample size from the entire working population of New York State provides us with a sample mean, a sample variance, and a sample standard deviation.

On the other hand, if we were able to ask every student in a school what their grade point average was and get an answer, this would be an example of a whole population. Using this information, we would be able to find the population mean, population variance, and population standard deviation.

Sample notation also differs from population notation, but don’t worry about these too much, because the formulas and meanings remain the same. For example, the population mean is represented by the greek letter, μ, but the sample mean is represented by x bar.

Now try calculating the standard deviation of each data set below on your own with the following practice problems!

Practice Questions:

Solutions:

Other Related Formulas

Mean Absolute Deviation (MAD):

The Mean Absolute Deviation otherwise known as MAD is another formula related to variance and standard deviation. In the MAD formula above, notice we are doing very similar steps, by finding the distance to the mean of each data point, only this time we are taking the aboslute value of the ditsance to the mean. Then we sum all the absolute value distances together and divide by the total number of data points.

Why do we use aboslute value in this formula? We take the absolut value, because if didn’t the distance to the means summed togther would cancel eachother out to get zero!

Where…

X = Data point value

μ = mean

N=Total number of data points

|X-μ|=absolute deviation

If we were to take the sample from our example earlier,60, 85, 95, 70, 80, in this post and find the MAD it would go something like this:

Expected Value:

Expected Value is the weighted average of all possible outcomes of one “game” or “gamble” based on the respective probabilities of each potential outcome for a discrete random variable. A “gamble” is defined by the following rules: 1) All possible outcomes are known 2) An outcome cannot be predicted 3) All possible outcomes are of numeric value and 4) The Game can be repeated multiple times under the same conditions.

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

Greetings math peeps! In today’s post, we are going to review some of the basics: combining like terms and distributive property. It’s so important to master the basics such as these, so you’re prepared and ready to handle the harder stuff that’s just around the corner, trust me they’re coming! And for those who already feel comfortable with these topics, great! Skip ahead and try the practice questions at the bottom of this post and happy calculating! 🙂

When do we combine “like terms?”

Combining like terms allows us to simplify and calculate our answer with terms that have the same variable and same exponent values only. For example, we can combine the following expression:

How do we combine like terms?

We add or subtract the whole number coefficients and keep the variable they have in common.

Why? We could not add these two terms together because their variables do not match! 2 is multiplied by x, while 3 is multiplied by the variable xy.

Why? We could not add these two terms together because their variables and exponents do not match! 2 is multiplied by x, while 3 is multiplied by the variable x^2 . Exponents for each variable must match to be considered like terms.

Distributive Property:

Combining like terms and the distributive property go hand in hand. The distributive property rule states the following:

There are no like terms to combine in the example above, but let’s see what it would like to use the distributive property and combine like terms at the same time with the following examples:

Example #1:

Example #2:

In some cases, we also have to distribute the -1 that can sometimes “hide” behind a parenthesis.

Try the following questions on your own on combining like terms and the distributive property and check out the video above for more!

Practice Questions:

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

Looking to review more of the basics? Check out this post on graphing equations of a line y=mx+b here.