Algebra 2/Trig.: Graphing Trig Functions

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 trigonometric functions to graph y=sin(x), y=cos(x), and y=tan(x). 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 Trig 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)?

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  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)?

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  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! 

How to Graph y=tan(x)?

Step 1: We are going to derive each degree value for tan 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  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! 🙂

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Algebra: Geometric Sequences

Hi everyone and welcome to Mathsux! In this post, we’re going to go over geometric sequences. We’ll see what geometric 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 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 15th 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:

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

Practice Questions:

  1. Find the 12th term given the following sequence: 1250, 625, 312.5, 156.25, 78.125, ….
  2. Find the 17th term given the following sequence: 3, 9, 27, 81, 243,…..
  3. Find the 10th term given the following sequence: 5000, 1250, 312.5, 78.125 …..
  4. 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!

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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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Algebra: Combining Like Terms and the Distributive Property

Greetings math peeps! In today’s post we are going to review some of the basics: combining like terms and the 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! 🙂

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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 2: Imaginary and Complex Numbers

Happy Wednesday and back to school season math friends! This post introduces imaginary and complex numbers when raised to any power exponent and when multiplied together as a binomial. When it comes to all types of learners, we got you between the video, blog post, and practice problems below. Happy calculating! 🙂

What are Imaginary Numbers?

Imaginary numbers happen when there is a negative under a radical and looks something like this:

Why does this work?

In math, we cannot have a negative under a radical because the number under the square root represents a number times itself, which will always give us a positive number.

Example:

But wait, there’s more:

When raised to a power, imaginary numbers can have the following different values:

Knowing these rules, we can evaluate imaginary numbers, that are raised to any value exponent! Take a look below:

-> We use long division, and divide our exponent value 54, by 4.

-> Now take the value of the remainder, which is 2, and replace our original exponent. Then evaluate the new value of the exponent based on our rules.

What are Complex Numbers?

Complex numbers combine imaginary numbers and real numbers within one expression in a+bi form. For example, (3+2i) is a complex number. Let’s evaluate a binomial multiplying two complex numbers together and see what happens:

-> There are several ways to multiply these complex numbers together. To make it easy, I’m going to show the Box method below:

Try mastering imaginary and complex numbers on your own with the questions below!

Practice:

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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