Central Angles Theorems: Geometry

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 angles theorems

Central Angle Theorems:

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

Central Angle Theorem #1:

central angles theorems

Central Angle Theorem #2:

central angles theorems

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

central angles theorems

Let’s do this one step at a time.

central angles theorems
central angles theorems
central angles theorems
central angles theorems
central angles theorems
central angles theorems

Practice Questions:

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 Intersecting Secants click this link here!

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TikTok Math Video Compilations

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! 🙂

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TikTok Math Video Compilations

Algebra:

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! 🙂

The Unit Circle: Algebra 2/Trig.

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.

The Unit Circle

Notice the following things about the unit circle above:

  1. Degrees follow a counter-clockwise pattern from 0 to 360 degrees.
  2. Values of cosine are represented by x-coordinates.
  3. Values of sine are represented by y-coordinates.
  4. 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:

The Unit Circle
The Unit Circle

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

The Unit Circle

Negative Degree Values:

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

The Unit Circle

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

The Unit Circle

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).

special triangles

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.

special triangles

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.

The Unit Circle

-> 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.

The Unit Circle

-> 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.

special triangles

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:

The Unit Circle

Solutions:

The Unit Circle

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! 🙂

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If you’re looking to learn about how to draw trigonometric functions, check out his post here!

Arithmetic Sequences: Algebra

Hi everyone and welcome to Mathsux! In this post, we’re going to go over arithmetic sequences. We’ll see what arithmetic sequences are, breakdown their formula, 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:

arithmetic sequences

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 123rd term?  That’s where the Arithmetic Sequence Formula comes in!

Arithmetic Sequence Formula:

arithmetic sequences

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

arithmetic sequence examples

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

arithmetic sequences

-> 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.

arithmetic sequence examples

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

arithmetic sequences

-> Now let’s write out our formula:

arithmetic sequences

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

Practice Questions:

  1. Find the 123rd term given the following sequence: 8, 12, 16, 20, 24, ….
  2. Find the 117th term given the following sequence: 2, 2.5, 3, 3.5, …..
  3. Find the 52nd term given the following sequence: 302, 300, 298, …..
  4. 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! 🙂

Also, if you’re looking to learn more about sequences, check out these posts on Geometric Sequences and Recursive Formulas!

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Geometry: 45º 45º 90º Special Triangles

45 45 90 triangle

Greetings math folks! In this post we are going to go over 45º 45º 90º special triangles and how to find the missing sides when given only one of its lengths. For even more examples, check out the video below and happy calculating! 🙂

Why is it “special”?

 The 45º 45º 90º triangle is special because it is an isosceles triangle, meaning it has two equal sides (marked in blue below).  If we know that the triangle has two equal lengths, we can find the value of the hypotenuse by using the Pythagorean Theorem.  Check it out below!

45 45 90 triangle

Now we can re-label our triangle, knowing the length of the hypotenuse in relation to the two equal legs. This creates a ratio that applies to all 45º 45º 90º triangles!

45 45 90 triangle

How do I use this ratio?

45 45 90 triangle

Knowing the above ratio, allows us to find any length of a 45º 45º 90º triangle, when given the value of one of its sides.

Let’s try an example:

45 45 90 triangle
45 45 90 triangle sides
45 45 90 triangle sides
45 45 90 triangle sides
45 45 90 triangle sides
45 45 90 triangle sides

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.

45 45 90 triangle sides
45 45 90 triangle formula
45 45 90 triangle formula
45 45 90 triangle formula
45 45 90 triangle formula
45 45 90 triangle formula
45 45 90 triangle formula

Now try mastering the art of the 45º 45º 90º special triangle on your own!

Practice Questions: Find the value of the missing sides.

45 45 90 triangle formula
45 45 90 triangle formula

Solutions:

45 45 90 triangle formula

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

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Synthetic Division and Factoring Polynomials: Algebra 2/Trig.

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.

Synthetic Division
Synthetic Division
Synthetic Division
Synthetic Division
Synthetic Division
Synthetic Division
Synthetic Division
Synthetic Division
Synthetic Division

Example #2: Factoring Polynomials

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

Synthetic Division

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)=2x3+4x2-10x-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! 🙂

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Variance and Standard Deviation: Statistics

Greetings math friends! In this post we’re going to go over variance and standard deviation. We will take this step by step to and explain the significance each have when it comes to a set of data. Get your calculators ready because this step by step although not hard, will take some serious number crunching! Check out the video below to see how to check your work using a calculator and happy calculating! 🙂

What is the Variance?

The variance represents the spread of data or 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?  This is what the variance tells us!

Don’t freak out but here’s the formula for variance, notated as sigma squared:

Variance and Standard Deviation

This translates to:

variance formula

Let’s try an example:

variance formula
Variance and Standard Deviation
Variance and Standard Deviation
Variance and Standard Deviation
Variance and Standard Deviation
Variance and Standard Deviation

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 formula for standard deviation happens to be very similar to the variance formula!

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

sample standard deviation

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 standard deviation

Now try calculating these statistics on your own with the following practice problems!

Practice Questions:

sample standard deviation

Solutions:

sample standard deviation

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

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Also! If you’re looking for more statistics, check out this post on how to create and analyze box and whisker plots here!

Transversals and Parallel Lines: Geometry

Happy Wednesday math friends! In this post we are going to look at transversals and parallel lines and find the oh so many congruent and supplementary angles they form when they come together! Congruent angles that form with these types of lines are more commonly known as Alternate Interior Angles, Alternate Exterior Angles, Corresponding angles, and Supplementary angles. Let’s look at this one step at a time:

What are Transversals and Parallel Lines?

When two parallel lines are cut by a diagonal line ( called a transversal) it looks something like this:

parallel lines and transversals

Each angle above has at least one congruent counterpart. There are several different types of congruent relationships that happen when a transversal cuts two parallel lines and we are going to break each down:

Transversals and parallel Lines

1) Alternate Interior Angles:

When a transversal line cuts across two parallel lines, opposite interior angles are congruent.

parallel lines and transversals

2) Alternate Exterior Angles:

When a transversal line cuts across two parallel lines, opposite exterior angles are congruent.

parallel lines and transversals

3) Corresponding Angles:

When a transversal line cuts across two parallel lines, corresponding angles are congruent.

Transversals and parallel Lines

4) Supplementary Angles:

Supplementary angles are a pair of angles that add to 180 degrees. 180 degrees is the value of distance found within a straight line, which is why you’ll find so many supplementary angles below:

parallel lines and transversals

Knowing the different sets of congruent and supplementary angles, we can easily find any missing angle values when faced with the following question:

Transversals and parallel Lines

-> Using our knowledge of congruent and supplementary angles we should be able to figure this out! Right away we can find angle 2 by noticing angle 1 and angle 2 are supplementary angles (add to 180 degrees). 

-> Knowing angle 2 is 50 degrees, we can now fill in the rest of our transversal angles based on our corresponding and supplementary rules.

Transversals and parallel Lines

Try the following transversal and parallel lines questions below! Some may a bit harder than the previous example, if you get stuck, check out the video that goes over a similar example above and happy calculating! 🙂

Practice Questions:

  1. Find the value of the missing angles given line r  is parallel to line  s and line t is a transversal. 
Transversals and parallel Lines

2. Find the value of the missing angles given line r is parallel to line s and line t is a transversal. 

Transversals and parallel Lines
Transversals and parallel Lines
Transversals and parallel Lines

Solutions:

Transversals and parallel Lines
Transversals and parallel Lines
Transversals and parallel Lines
Transversals and parallel Lines

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

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Also, if you’re looking to learn about the difference between parallel and perpendicular lines, check out this post here!

Box and Whisker Plots, IQR and Outliers: Statistics

Ahoy math friends! This post takes a look at one method of analyzing data; box and whisker plots. Box and whisker plots are great for visually identifying outliers and the overall spread of numbers in a data set. We will go over step by step how to create a box and whisker plot given a set of data, we will then look at how to find the interquartile range and upper and lower outliers. If you have any questions, don’t hesitate to check out the video or comment below. Stay curious and happy calculating! 🙂

Looking for more MathSux? Check out this post on variance and standard deviation here!

Box plots look something like this:

Screen Shot 2020-09-02 at 11.19.22 AM.png

Why Box Plots?

Box Plots are a great way to visually see the distribution of a set of data.  For example, if we wanted to visualize the wide range of temperatures found in a day in NYC, we would get all of our data (temperatures for the day), and once a box plot was made, we could easily identify the highest and lowest temperatures in relation to its median (median: aka middle number).

From looking at a Box Plot we can also quickly find the Interquartile Range and upper and lower Outliers. Don’t worry,  we’ll go over each of these later, but first, let’s construct our Box Plot!

Screen Shot 2020-09-02 at 11.20.42 AM
Screen Shot 2020-09-02 at 11.21.28 AM.png
Box and Whisker Plots

->  First, we want to put all of our temperatures in order from smallest to largest.
-> Now we can find Quartile 1 (Q1), Quartile 2 (Q2) (which is also the median), and Quartile 3 (Q3).  We do this by splitting the data into sections and finding the middle value of each section.

Q1=Median of first half of data

Q2=Median of entire data set

Q3=Median of second half of data

-> Now that we have all of our quartiles, we can make our Box Plot! Something we also have to take notice of, is the minimum and maximum values of our data, which are 65 and 92 respectively. Let’s lay out all of our data below and then build our box plot:

Box and Whisker Plots
Box and Whisker Plots

Now that we have our Box Plot, we can easily find the Interquartile Range and upper/lower Outliers.

Screen Shot 2020-09-05 at 11.21.54 PM

->The Interquartile Range is the difference between Q3 and Q1. Since we know both of these values, this should be easy!

Screen Shot 2020-09-05 at 11.22.02 PM

Next, we calculate the upper/lower Outliers.

Box and Whisker Plots

-> The Upper/Lower Outliers are extreme data points that can skew the data affecting the distribution and our impression of the numbers. To see if there are any outliers in our data we use the following formulas for extreme data points below and above the central data points.

Screen Shot 2020-09-05 at 11.24.27 PM

*These numbers tell us if there are any data points below 44.75 or above 114.75, these temperatures would be considered outliers, ultimately skewing our data. For example, if we had a temperature of  Screen Shot 2020-09-05 at 11.26.38 PMor Screen Shot 2020-09-05 at 11.29.25 PM these would both be considered outliers.

Screen Shot 2020-09-05 at 11.24.35 PM

Practice Questions:

Screen Shot 2020-09-05 at 11.34.21 PM

Solutions:

Screen Shot 2020-09-05 at 11.37.06 PM
Box and Whisker Plots
Screen Shot 2020-09-05 at 11.38.10 PM
Box and Whisker Plots

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

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How to Construct an Equilateral Triangle?: Geometry

Happy Wednesday math peeps! This post introduces constructions by showing us how to construct an equilateral triangle by using a compass and a ruler. For anyone new to constructions, this is the perfect topic for art aficionados since there is more drawing than there is actual math. 

What is an Equilateral Triangle?

Equilateral Triangle: A triangle with three equal sides.  Not an easy one to forget, the equilateral triangle is super easy to construct given the right tools (compass+ ruler). Take a look below:

how to construct an equilateral triangle

Now, for our Example:

how to construct an equilateral triangle

Solution:

How to Construct an Equilateral Triangle

What’s Happening in this GIF? 

1. Using a compass, measure out the distance of line segment  Screen Shot 2020-08-25 at 4.19.02 PM.

 2. With the compass on point A, draw an arc that has the same distance as Screen Shot 2020-08-25 at 4.19.02 PM.

 3. With the compass on point B, draw an arc that has the same distance as Screen Shot 2020-08-25 at 4.19.02 PM.

4. Notice where the arcs intersect? Using a ruler, connect points A and B to the new point of intersection. This will create two new equal sides of our triangle!

Still got questions? No problem! Don’t hesitate to comment with any questions. Happy calculating! 🙂

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Looking to learn more about triangles? Check out this post on right triangle trigonometry here!