Inscribed Angles & Intercepted Arcs: Geometry

Ahoy math friends and welcome to MathSux! In this post, we are going to go over the inscribed angle theorem which dives into inscribed angles and intercepted arcs. We’ll break down the main basic rule for inscribed angle theorem and the three inscribed angle theorems associated with this rule. If you are looking for more circle theorems, check out these posts on the Intersecting Secants Theorem and Central Angles Theorem. Also, don’t forget to check out the video and practice questions to truly master the topic below. Happy calculating! 🙂

Inscribed Angles:

When two chords come together to touch the outline of a circle, they create something called an inscribed angle. Unlike central angles, an inscribed angle is equal to half the measure of the arc length. In the example below, we can see that the inscribed angle ACB is equal to 25º while its corresponding intercepted arc AB is twice the inscribed angle at 50º. Knowing this allows us to find the value of unknown arcs or inscribed angles.

Inscribed Angles & Intercepted Arcs

Inscribed Angle Theorem:

There are three parts to the inscribed angle theorem to know based on the rule stated above, check them out below!

The inscribed angle theorem states that, in a circle when two inscribed angles intercept the same arc, the angles are congruent. Notice below how angle A and angle B both have the same arc CD (hi-lighted in pink). In this theorem, we have two inscribed angles, angle A and angle B, that intercept the same arc, arc CD.

Inscribed Angles & Intercepted Arcs

Theorem #2: In a circle when an angle is inscribed by a semicircle, it forms a  90º angle (or a right angle). If you look at angle BAC below, you’ll notice that the arc it corresponds with is a semicircle and that angle BAC forms a right angle.

Theorem #3: When a quadrilateral is inscribed within a circle, the opposite angles formed are supplementary, meaning that they add to 180º. The proof below shows angles A and C (hi-lighted in green) as supplementary, but this proof would also work for opposite angles B and D (hi-lighted in pink).

Inscribed Angles & Intercepted Arcs

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

a) Step 1: To find the value of angle CDB we need to look at our given information. We know that angle CAB=85º, notice that this follows theorem number 3, “When a quadrilateral is inscribed in a circle, opposite angles are supplementary.” Therefore, we must subtract 110º from 180º to find the value of angle CDB.

b) Step 2: For finding angle ABD, we’re going to use the same theorem we used in part a, opposite supplementary angles of an inscribed quadrilateral are supplementary.

c) Step 3: Next, to find the value of arc ABD, we need to use the basic inscribed angle theorem that tells us an inscribed angle is equal to half the measure of its arc. Then use some basic algebra to solve for arc ABD.

d) Step 4: To find arc ACD, we need to use the basic inscribed angle theorem that tells us an inscribed angle is equal to the value of its arc, then use algebra to solve similar to part c.

If this looks confusing, check out the video above! And when you are ready master this topic with the practice questions below!

Practice Questions:

Use the inscribed angle theorem outlined above to answer the following questions.

Solutions:

Still got questions?  No problem! Check out the video above or comment below for any questions and follow for the latest MathSux posts. Happy calculating! 🙂

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Looking for more posts on Circle Theorems? Check out the posts below!

Circle Theorems & Formulas

Central Angle Theorem

Intersecting Secants Theorem

Area of a Sector

Circle Theorems

Summation Notation: Algebra 2

Hi everyone and welcome to MathSux! In this post we are going to go over summation notation (aka sigma notation). The summing of a series isn’t hard as long as you know how to read the notation! We will go over an example and breakdown what each part of this notation represents step by step. When you are ready, please don’t forget to check out the practice questions at the end of this post to truly master the topic. Thanks for stopping by and happy calculating! 🙂

What is Summation Notation?

Summation notation lets us write a series in an easy and short-handed way.  Before we go any further we also need to define a series!

Series: The sum of adding each term within an infinite sequence. This can include arithmetic or geometric sequences we are already familiar with. For example, let’s say we have the arithmetic sequence: 2,4,6,8, ….. now with a series we are adding all of these terms together: 2+4+6+8+……

Now back to summations. Summations allow us to quickly understand that the sequence being added together is done so on an infinite or finite basis by giving us a range of values for which the unknown variable can be evaluated and summed together.  Summation notation is represented with the capital Greek letter sigma, Σ, with numbers below and above as limits for calculation and the series that must be evaluated to the right.

If this sounds confusing, don’t worry, it might sound more confusing than it actually is! Take a look at the breakdown for sigma notation below:

Summation Notation

Wait, what does the above summation say?

Translation: It tells us to evaluate the expression, n+1 by plugging in 1 for n, 2 for n, and 3 for n and then wants us to sum all three solutions together.

Take a look below to see how to solve this step by step:

Summation Notation

Check out the video above to see more examples step by step! When you’re ready to try them on your own, check out the practice problems below:

Practice Questions:

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions below. Thanks for stopping by and happy calculating! 🙂

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Looking for something similar to sigma notation? Check out this post on geometric sequences here! If you’re looking for more statistics formulas, check out how to find expected value!

Continuously Compounding Interest Formula

A=Pert

P=Principal (Original $ Amount Invested)

e=Mathematical Constant

r=Interest Rate

t=Time

Hi everyone and welcome to MathSux! In this post, we are going to go over the continuously compounding interest formula, A=Pert! This is a great topic as it relates to finance and real-world money situations (sort of). We are going to break down everything step by step by understanding what the continuously compounding interest rate formula A=Pert is, identifying each component of the formula, and then applying it to an example. If you’re looking for more, don’t forget to check out the video and the practice questions at the end of this post. Happy calculating! 🙂

What is Continuous Compounding Interest Formula?

Let’s say we have $500 and we want to invest it.

What if it compounded interest once a year? 

Twice a year? 

Once a day, or 365 days a year?

What if we compounded interest every second of the day for a total of 86,400 seconds throughout the year!?

And what if we kept going, making the number of times compounded annually more and more often to occur every half second? This is what Continuous Compounding Interest is, and it tells us how much we earn on a principle (original amount) if the compound interest rate for the year were to be granted an infinite number of times.

The weird thing is that continuous compounding interest is technically impossible (I’ve yet to see a bank that offers an infinite number of compounding interest!).  Even though it is impossible, in math and finance, we look at continuous compounding interest for theoretical purposes, in other words, it’s for money nerds! Luckily, it comes with an easy-to-use formula, let’s take a look:

Continuously Compounding Interest Formula

Now, let’s see this formula in action with the following Example:

Step 1: First, let’s write out our formula and identify what each value represents based on the question.

Continuously Compounding Interest Formula

Step 2: Fill in our formula with the given values and solve.

Practice Questions:

1) Sally invested $1000 which was then continuously compounded by 4%. How much money will Sally have after 5 years?

2) Brad invested $1500 into an account continuously compounded by 5%. How much money will he have after 7 years?

3) Fran invests $2000 into an account that is continuously compounded by 1%. How much money will Fran earn by year 5?

Solutions:

1) $1,221.40

2) $2,128.60

3) $2,102.54

Want to make math suck just a little bit less? Subscribe to my Youtube channel for free math videos every week! 🙂

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Looking to review more topics in Algebra? Check our Algebra page here!

NumWorks Calculator Review

Greeting math friends and welcome to MathSux! In today’s post we are going to review and take a look at how to use the graphing calculator available by the French company, NumWorks.

In this NumWorks calculator review, first impressions are that this is a serious competitor for Texas Instruments and offers more features than a typical calculator with a focus on statistics, data analysis, and even computer programming! Check out the video below to see the un-boxing, full review, and how to use this calculator step by step. Happy calculating! 🙂

NumWorks Calculator Stand Out Features:

NumWorks Calculator Review

1) The Home Screen: Works and looks like apps on an iPhone. It is super easy to use, and includes apps such as the regular graphing calculator we’re all used to, as well as, Python, Statistics, Probability, Equation Solver, Sequences, and Regression.

2) The Equation Solver: Punch in any function and find it’s x-values and discriminant! Very cool!

3) Python: Yes, this calculator is programmable via Python! It also includes pre-made scripts that you can easily run. This is great for aspiring programmers and important for today’s economy.

4) Exam Mode: Teachers can make students put their calculators in exam mode and watch their students calculators light up in red to prove there’s no cheating funny business going on! Warning though, this will delete all of your data including the pre-made Python scripts. But you can always hit the reset button in the back to reset.

NumWorks Calculator Review

Did I mention math teacher’s can potentially get a free calculator from NumWorks? Check out the link here!

Has anyone else tried this graphing calculator from NumWorks? What were your first thoughts? Let me know in the comments and happy calculating!

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For more math resources, check out this post here and happy calculating! 🙂

Square Inscribed in a Circle Construction

Greetings math friends and welcome to MathSux! In this week’s post, we are going to take a step-by-step look at how a square inscribed in a circle construction works! Within this post we have a video, a GIF, and a step-by-step written explanation below, the choice of learning how to do this construction is up to you! Happy Calculating! 🙂

Square Inscribed in a Circle Construction

How to Construct a Square Inscribed in Circle:

Step 1: First, we are going to draw a circle using a compass (any size).

Step 2: Using a ruler, draw a diameter or straight line across the length of the circle, going through its midpoint.

Step 3: Next, open up the compass across the circle. Then take the point of the compass to one end of the diameter and swing the compass above the circle, creating an arc.

Step 4: Keeping that same length of the compass, go to the other side of the diameter and swing above the circle again making another arc until the two arcs intersect.

Step 5: Now, we are going to repeat steps 3 and 4, this time creating arcs below the circle using the same compass size.

Step 6: Connect the point of intersection above and below the circle using a ruler or straight edge. This creates a perpendicular bisector, cutting the diameter in half and forming a 90º angle.

Step 7: Lastly, we are going to use a ruler to connect each corner point to one another creating a square!

Constructions and Related Posts:

Looking to construct more than just a square inside a circle? Check out these related posts and step-by-step tutorials on geometry constructions below!

Construct an Equilateral Triangle

Perpendicular Line Segment through a Point

Angle Bisector

Construct a 45º angle

Altitudes of a Triangle (Acute, Obtuse, Right)

How to Construct a Parallel Line

Bisect a Line Segment

Best Geometry Tools!

Looking to get the best construction tools? Any compass and straight-edge will do the trick, but personally, I prefer to use my favorite mini math toolbox from Staedler. Stadler has a geometry math set that comes with a mini ruler, compass, protractor, and eraser in a nice travel-sized pack that is perfect for students on the go and for keeping everything organized….did I mention it’s only $7.99 on Amazon?! This is the same set I use for every construction video in this post. Check out the link below and let me know what you think!

Still got questions? No problem! Don’t hesitate to comment with any questions or concerns below. Also if you’re looking for a different type of construction you don’t yet see here, please let me know! Happy calculating! 🙂

Let’s be friends! Check us out on the following social media platforms for even more free MathSux practice questions and videos:

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Looking for something similar to square inscribed in a circle construction? Check out this post here on how to construct and equilateral triangle here! And if you’re looking for even more geometry constructions, check out the link here!

How to Calculate Z-Score?: Statistics

Hi there and welcome to MathSux! In this post, we are going to explore how to calculate z-score and the normal distribution. We’ll do this by examining the normal curve and learning how to find probability finding z-score and using the mean, standard deviation, and specific data points. Fore more info and more MathSix don’t forget to check out the video and practice questions below. Happy calculating!

What is a Normal Curve?

A normal curve is a bell shaped curve that shows the distribution of data evenly spread with respect to the mean. If you look at the normal curve below, the area under the curve shows all the possible probabilities of a certain data point occurring, notice the curve is higher towards the center mean, μ, and gets smaller as the distance from μ grows. The distance from μ is measured by the standard deviation, a unique unit of measurement that is specific to each group of data.

Mean: The mean always falls directly in the center of our normal curve. It is the average of our data, and always falls right in the middle.

Standard Deviation:  This value is used as a standard unit of measurement for the data, measuring the distance between each data point in relation to the mean throughout the entire data set. For a review on what standard deviation is and how to calculate it, check out this post here.

Now for our normal curve:

Notice half of the data is below the mean, μ, while the other half is above? The normal curve is symmetrical about the mean, μ!

How to Calculate Z-Score?

Z Score can tell us at what percentile a certain point in the data set falls in relation to the rest of the mean by using the standard deviation as a unit of measurement.  If this sounds confusing, it’s ok! Take a look at the following formula:

How to Calculate Z-Score?

We use the above formula in conjunction with  a z table which tells us the probability under the curve for a certain point.

Solution:

a) What percent of student scored below 500?

Step 1: First, let’s draw out our given information the mean=500, standard deviation=100, and the data point the question is asking for x=500 onto a normal curve. Notice that we want to find the value of the area under the curve shaded in pink.  This will tell us the percent of students that scored below 500.

How to Calculate Z-Score?

Step 2: We need to find the z-score by, using the data point given to us x=500, the mean=500, and the standard deviation, sigma=100.

Step 3: Yes, we have a zero! Now we need to take our z table and line up our chart. Notice that the chart finds the probability for everything at the beginning of the normal curve and on.  This is perfect for answering our question!

How to Calculate Z-Score?

Step 4: The table gives us our solution of .5000.  If we multiply .5000 times 100 it gives us the percent of students who scored below 500 at 50%.

b) What percent of student scored above 620?

Step 1: First, let’s draw out our given information the mean=500, standard deviation=100, and the data point the question is asking for x=620 onto a normal curve. Notice that we want to find the value of the area under the curve shaded in pink.  This will tell us the percent of students that scored above 620.

How to Calculate Z-Score?

Step 2: We need to find the z-score by, using the data point given to us x=620, the mean=500, and the standard deviation, sigma=100.

How to Calculate Z-Score?

Step 3: Yes, we got 1.2! Now we need to take our z table and line up our chart. Notice that the chart finds the probability for everything at the beginning of the normal curve and on.  This is means to find the percent we are looking for, we need to subtract our answer from one since we want the value of probability on the right side of the curve (the z-table only provides the left side).

How to Calculate Z-Score?

Step 4: The table gives us our solution of .8849.  If we subtract this value from 1 then multiply that value times 100 it gives us the percent of students who scored above 620.

How to Calculate Z-Score?

C) What is the highest score a student could receive if the students was in the 16.11th percentile?

Step 1: In this question we have to work backwards by first identifying, where on the z-score table is the number .1611 and then filling in our z score formula to find x, the missing data point (in this case test score).

Search the table for .1611:

How to Calculate Z-Score?

Notice that .1611 can be found on the z-table above with z-score -0.99.  This is what we’ll use to find the unknown data point!

Step 2: We need to find the unknown test score by, using the z score we just found z=-0.99, the mean=500, and the standard deviation, sigma=100.

How to Calculate Z-Score?

Step 3: Solve for x.

How to Calculate Z-Score?

Practice Questions:

The grades on a final English exam are normally distributed with a mean of 75 and a standard deviation of 10.

a) What percent of students scored below a 60?

b) What percent of students scored above an 89?

c) What is the highest possible grade that included in the 4.46th percentile?

d) What percent of students got at least a 77?

Solutions:

a) 6.68%

b) 8.08%

c) 58

d) 42.07%

Want to make math suck just a little bit less? Subscribe to my Youtube channel for free math videos every week! 🙂

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Looking for more Statistics?

Difference between Bar Graphs and Histograms

Box and Whisker Plots

Variance and Standard Deviation

Expected Value

Simultaneous Equations: Algebra

Happy new year and welcome to Math Sux! In this post we are going to dive right into simultaneous equations and how to solve them three different ways! We will go over how to solve simultaneous equations using the (1) Substitution Method (2) Elimination Method and (3) Graphing Method. Each and every method leading us to the same exact answer! At the end of this post don’t forget to try the practice questions choosing the method that best works for you! Happy calculating! 🙂

What are Simultaneous Equations?

Simultaneous Equations are when two equations are graphed on a coordinate plane and they intersect at, at least one point.  The coordinate point of intersection for both equation is the answer we are trying to find when solving for simultaneous equations. There are three different methods for finding this answer:

We’re going to go over each method for solving simultaneous equations step by step with the example below:

Method #1: Substitution

The idea behind Substitution, is to solve for 1 variable first algebraically, and the plug this value back into the other equation solving for one variable.  Then solving for the remaining variable.  If this sounds confusing, don’t worry! We’re going to do this step by step:

Step 1: Let’s choose the first equation and move our terms around to solve for y.

Simultaneous Equations

Step 3: The equation is set up and ready to solve for x!

Simultaneous Equations

Step 4: All we need to do now, is plug x=3 into one of our original equations to solve for y.

Simultaneous Equations

Step 5: Now that we have solved for both x and y, we have officially found where these two simultaneous equations meet!

Simultaneous Equations

Method #2: Elimination

The main idea of Elimination is to add our two equations together to cancel out one of the variables, allowing us to solve for the remaining variable.  We do this by lining up both equations one on top of the other and adding them together.  If variables at first do not easily cancel out, we then multiply one of the equations by a number so it can. Check out how it’s done step by step below!

Step 1: First, let’s stack both equations one on top of the other to see if we can cancel anything out:

Simultaneous Equations

Step 2: Our goal is to get a 2 in front of y in the first equation, so we are going to multiply the entire first equation by 2.

Simultaneous Equations

Step 3: Now that we multiplied the entire first equation by 2, we can line up our two equations again, adding them together, this time canceling out the variable y to solve for x.

Simultaneous Equations

Step 4: Now, that we’ve found the value of variable x=3, we can plug this into one of our equations and solve for missing unknown variable y.

Simultaneous Equations

Step 5: Now that we have solved for both x and y, we have officially found where these two simultaneous equations meet!

This image has an empty alt attribute; its file name is Screen-Shot-2020-12-30-at-10.21.46-AM.png

Method #3: Graphing

The main idea of Graphing is to graph each a equation on a coordinate plane and then see at what point they intersect.  This is the best method to visualize and check our answer!

Step 1: Before we start graphing let’s convert each equation into y=mx+b (equation of a line) form.

Equation 1:

Equation 2:

Step 2: Now, let’s graph each line, y=3x-4 and y=-x+8, to see at what coordinate point they intersect.

Simultaneous Equations

Need to review how to draw an equation of a line? Check out this post here! Notice we got the same exact answer using all three methods (1) Substitution (2) Elimination and (3) Graphing.

Ready to try the practice problems on your own?! Check them out below!

Practice Questions:

Solve the following simultaneous equations for x and y.

Solutions:

  1. (1, 3)
  2. (4,5)
  3. (-1, -6)
  4. (3, -3)

Want more MathSux?  Don’t forget to check out our Youtube channel and more below! And if you have any questions, please don’t hesitate to comment below. Happy Calculating!

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If you are looking for a challenge, try solving three equations with three unknown variables, in this post here!

30 60 90 Triangle

Hi everyone and welcome to MathSux! In this post, we are going to break down 30 60 90 degree special right triangles. What is it? Where did it come from? What are the ratios of its side lengths and how do we use them? You will find all of the answers to these questions about this right angled triangle in this post. Also, don’t forget to check out the video below and practice questions at the end of this post to truly become a 30 60 90 special right triangle master! Happy calculating! 🙂

If you want to learn about the other special triangle, 45 45 90 triangle, check out this post here. And if you’re looking to make math suck just a little bit less? Subscribe to our Youtube channel for free math videos every week! 🙂

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30 60 90 Triangle Ratio:

30 60 90 triangle side lengths

Notice that the 30 60 90 triangle is made up of one right angle across from the hypotenuse (which is always going to be the longest side), a 60 degree angle with a longer leg on the opposite side, and a 30 degree angle measure across from the shorter leg.

What is a 30 60 90 Triangle and why is it “Special”?

The 30 60 90 triangle is a special right triangle because it forms an equilateral triangle when a mirror image of itself is drawn. This means that all sides are equal when a mirror image of the triangle is drawn which allows us to find the ratio between each of the sides of the triangle by using the Pythagorean Theorem. Check it out below!

30 60 90 Special Triangles

Now let’s draw a mirror image of our triangle to create an equilateral triangle.  Next, we can label the length of the new side opposite 30 degrees (the shorter leg) “a,” and add this new mirror image length with the original we had to get, a total of a+a=2a.

30 60 90 Special Triangles
30 60 90 Special Triangles

If we look at our original 30 60 90 triangle, we now have the following values for each side based on our equilateral triangle. Notice we still need to find the length value of the longer leg, opposite angle 60 degrees.

30 60 90 Special Triangles
Pythagorean Theorem Formula
Fill in values from our triangle longer leg
Distribute the exponent and start solving for our “?” by subtracting a^2 from both sides.
Now take the square root of both sides to solve for “?.”	We have found a solution! Our missing side length is equal to a√3.

To sum up, we have applied the Pythagorean Theorem formula, filled in values found in our triangle for each length, distributed the exponent and subtracted a2 from both sides, took the square root, and finally found our solution for our ratio for the longer leg opposite 60 degrees.

Now we can re-label our triangle, knowing the length of the hypotenuse in relation to the two legs. This creates a ratio that applies to all 30 60 90 triangles!

30 60 90 triangle side lengths

How do I use this ratio?

30 60 90 triangle side lengths

Knowing the above ratio, allows us to find any length of any and every 30 60 90 triangle, when given the value of one of its sides. Let’s tsee how this ratio works with some examples.

Example #1:

30 60 90 triangle side lengths

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

30 60 90 triangle side lengths

Step 2: Notice we are given the value of a, which is equal to 4, knowing this we can now fill in each length of our triangle based on the ratio of a 30 60 90 triangle.

30 60 90 triangle side lengths
30 60 90 triangle side lengths

Now let’s look at another Example where we are given the length of the hypotenuse and need to find the values of the other two missing sides.

Example #2:

30 60 90 triangle side lengths

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

30 60 90 triangle side lengths

Step 2: Notice we are given the value of the hypotenuse, 2a=20. Knowing this we can find the value of a by dividing 20 by 2 to get a=10. Once we have the value of a=10, we can easily find the length of the last, longer leg based on the 30 60 90 ratio:

30 60 90 triangle side lengths
30 60 90 triangle side lengths

Now for our last Example, we will see how to find the value of the shorter leg and hypotenuse, when we are given the side length of the longer leg across from 60º and need to find the other two missing sides.

Example #3:

30 60 90 triangle side lengths

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

30 60 90 triangle side lengths

Step 2: In this case, we need to use little algebra to find the value of a, using the ratio for 30 60 90 triangles.

30 60 90 triangle side lengths
We set up this equation, knowing that 9 is across from 60 degrees, and that a√3=9. 	To solve for a, we divide square root both sides by √3. 	We get a solution a=9/√3! Yay! We cannot have a radical in the denominator, so we rationalize the denominator by multiplying the numerator and denominator by √3.

Now that we have one piece of the puzzle, the value of a, let’s fill in the value of the shorter leg of our triangle below:

Finally, let’s find the value of the length of the hypotenuse, which is equal to 2a in our ratio. Knowing the length of all sides, we can fill in the lengths of each side of our triangle for our solution.

Think you are ready to master these types of questions on your own? Try the practice problems below!

Practice Questions:

Find the value of the missing sides of each 30 60 90 degree triangle.

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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Related Trigonometry Posts:

The Unit Circle

Basic Right Triangle Trigonometric Ratios (SOH CAH TOA)

4545 90 Special Triangles

Factoring Trigonometric Functions

Graphing Trigonometric Functions

Trig Identities

Transforming Trig Functions

Law of Cosines

Law of Sines

NYE Ball Fun Facts: Volume & Combinations

Greetings math friends, and Happy New Year! In today’s post we’re going do something a little different and take a look at the math behind the very famous and very shiny New Year’s Eve Ball that drops down every year at midnight.  We’ll break down the shape, the volume, and the number of those dazzling Waterford crystals (and no this post isn’t sponsored) and look at some NYE Ball Fun Facts.

NYE Ball Fun Facts

Shape: Geodesic Sphere

Yes, apparently the shape of the New Year’s ball is officially called a “Geodesic Sphere.”  It is 12 feet in diameter and weighs 11,875 pounds.

Volume (Estimate): 288π ft3

If we wanted to estimate the volume of the New Year’s Ball we would could use the formula for volume of a sphere:

NYE Ball Fun Facts

Number of Waterford Crystals: 2,688

Talk about the ultimate shiny bauble! The NYE ball lights up the night with all 2,688 crystals in the shape of different sized triangles, each with heights of 5.75 inches or 4.75 inches.

NYE Ball Fun Facts

Number of Lights: 48 light emitting diodes (LED’s)

On each triangle, there are 48 LEDs: 12 red, 12 blue, 12 green, and 12 white, for a total of 32,256 LEDs on the entire NYE ball itself.

NYE Ball Fun Facts

Permutations and Combinations:

Permutations: With this many lights and colors, there are over a billion potential permutations of colors on the entire NYE ball.

Combinations: Let’s break down one triangle with 48 LED lights each with 12 red, 12 blue, 12 green, and 12 white LEDs. How many possible combinations of lights are possible if we were to choose 7 blue, 5 red, 10 green, and 1 white turned on all at the exact same time?

We end up with the combination formula below:

NYE Ball Fun Facts

That means that there are 496,793,088 possible ways that 7 blue lights, 5 red lights, 10 green lights, and 1 white light can be lit up on a triangle that is part of the entire NYE ball!

Interested in more NYE fun facts?  Check out the sources of this article here.

NYE Fact Sheet from: timessquarenyc.org

NYE Ball picture: Timesquareball.net

If you like finding the volume of the NYE ball maybe, you’ll want to find the volume of the Hudson Yards Vessal in NYC here.  Happy calculating and Happy New Year from MathSux!

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Transforming Trig Functions: Amplitude, Frequency, Period, Phase Shifts

Hi everyone, and welcome to MathSux! In this post, we are going to break down transforming trig functions by identifying their amplitude, frequency, period, horizontal phase shift, and vertical phase shifts. Fear not! Because we will break down what each of these terms (amplitude, frequency, period, horizontal phase shift and vertical phase shifts) mean and how to find them when looking at a trigonometric function and then apply each of these changes step by step to our graph. In this particular post, we will be transforming and focusing on a cosine function, but keep in mind that the same rules apply for transforming sine functions as well (example shown below in practice).

And if you’re ready for more, check out the video and the practice problems below, happy calculating! 🙂

*For a review on how to derive the basic Trig functions (y=sinx, y=cosx, and y=tanx), click here.

What are the Different Parts of a Trig Function?

When transforming trig functions, there are several things to look out for, let’s take a look at what each part of a trig function represents below:

Trig functions

Amplitude: The distance (or absolute value) between the x axis and the highest point on the graph.

Frequency: This is the number of cycles that happen between 0 and 2π. (Α “cycle” in this case is the number of “s” cycles for the sine function).

Period: The x-value/length of one cycle. (Α “cycle” in this case is the number of “s” cycles for the sine function). This is found by looking at the graph and seeing where the first cycle ends, or, by using the formula:       

Horizontal Shift: When a trigonometric function is moved either left or right along the x axis.

Vertical Shift: When a trigonometric function is moved either up or down along the y axis.

Let’s try an Example, by graphing the following trig function step by step by identifying the amplitude, frequency, period, vertical shift, and horizontal phase shift.

Step 1: First let’s label and identify all the different parts of our trig function and what each part represents.

Trig functions

Step 2: Now let’s transform our trig graph one step at a time.  First, let’s start graphing y=cos(x) without any transformations, the basic graph.

Trig functions
Graph y=cos(x)

Step 3: Next, let’s add our amplitude of 2, otherwise known as the height, or distance to the x-axis.  To do this our highest and lowest points on the y-axis will now be moved to 2 and -2 respectively. 

Trig functions
Re-draw y=cos(x), with an amplitude of 2 to get y=2cos(x)

Step 4: Next, we can apply our horizontal shift to the left by (π/2) or 90º.  To do this, we need to look at where negative (π/2) is on our graph at (-π/2) and move our entire graph over to start at this new point, “shifting” over each coordinate point by (π/2) along the x axis.

Trig functions
Shift our graph y=2cos(x) over by 90º to the left to get y=2cos(x+(π/2))

Step 5: For our last transformation, we have a vertical phase shift up 1 unit.  All this means is that we are going to shift our entire graph up by 1 unit along the y axis.

Trig function graph
phase shift
vertical shift
Shift our entire graph y=2cos(x+(π/2)) up one unit along the y-axis to get y=2cos(x+(π/2))+1

Think you are ready to try graphing trig functions and identifying the amplitude, frequency, period, vertical phase shift, and horizontal phase shift? Check out the practice questions and answers below!

Practice Questions:

When you’re ready check out the function transformations solutions below:

Solutions:

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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