Trigonometry: Basics of the Unit Circle

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:

  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:

Now let’s look at some examples on how to find trigonometric functions using the unit 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! 🙂

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Algebra: Arithmetic Sequences

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:

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:

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:

  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! 🙂

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Geometry: 45º 45º 90º Special Triangles

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!

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!

How do I use this ratio?

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:

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.

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

Practice Questions: Find the value of the missing sides.

Solutions:

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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Algebra 2: Synthetic Division and Factoring Polynomials

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.

When can we use Synthetic Division?

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.

Check out the Example below to see synthetic division in action:

Synthetic Division can also be used when Factoring Polynomials!

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

Check!

Try the practice problems on your own below!

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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Algebra: Variance and Standard Deviation

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:

This translates to:

Let’s try an example:

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:

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:

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

Practice Questions:

Solutions:

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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Geometry: Transversals and Parallel Lines

Happy Wednesday math friends! In this post we are going to look at parallel lines and transversals 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?

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

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:

1) Alternate Interior Angles:

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

2) Alternate Exterior Angles:

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

3) Corresponding Angles:

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

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:

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

-> 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.

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. 

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

Solutions:

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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Algebra: Box Plots, Interquartile Range and Outliers, Explained!

Ahoy math friends! This post takes a look at one method of analyzing data; the box plot method. This method is great for visually identifying outliers and the overall spread of numbers in a data set.

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->  First, we want to put all of our temperatures in order from smallest to largest.
Screen Shot 2020-09-02 at 11.21.28 AM.png-> 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.Screen Shot 2020-09-05 at 11.19.22 PM

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:

Screen Shot 2020-09-05 at 11.19.27 PM

Screen Shot 2020-09-05 at 11.20.45 PM

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 PMNext, we calculate the upper/lower Outliers.

Screen Shot 2020-09-05 at 11.23.45 PM

-> 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 PMSolutions:

Screen Shot 2020-09-05 at 11.37.06 PM

Screen Shot 2020-09-05 at 11.37.39 PM

Screen Shot 2020-09-05 at 11.38.10 PM

Screen Shot 2020-09-05 at 11.39.06 PM

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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Geometry: How to Construct an Equilateral Triangle?

 

Happy Wednesday math peeps! This post introduces constructions by showing 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.  Screen Shot 2020-08-25 at 4.09.58 PM.pngEquilateral 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:Screen Shot 2020-08-25 at 3.56.17 PM.png

Solution:

Construction-GIF-v2

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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Algebra: Completing the Square

Learn how to Complete the Square by clicking on the Youtube video and trying the practice problems below. Happy Calculating! 🙂

Click the picture below to view the Youtube video.

Complete the Square copy

Screen Shot 2020-05-23 at 5.28.18 PMPractice Questions:

Screen Shot 2020-05-23 at 5.28.54 PM

Solutions:

Screen Shot 2020-05-23 at 5.29.19 PM

Need more of an explanation?  Check out why we complete the square in the first place here and please don’t forget to subscribe! 🙂