How to use SOH CAH TOA: Geometry

Welcome back to Mathsux! This week, we’re going to go over how to find missing angles and side lengths of right triangles by using trigonometric ratios (sine, cosine, and tangent) (aka how to use SOH CAH TOA).  Woo hoo! These are the basics of right triangle trigonometry, and provides the base for mastering so many more interesting things in trigonometry! So, let’s get to it!

Also, if you have any questions about anything here, don’t hesitate to comment below or watch the video below. Also, don;r forget to subscribe to Math Sux for FREE math videos, lessons, and practice questions every week. Happy calculating! 🙂

How to use SOH CAH TOA?

Trigonometric Ratios (more commonly known as Sine, Cosine, and Tangent) are ratios that naturally exist within a right triangle.  This means that the sides and angles of a right triangle are in proportion within itself.  It also means that if we are missing a side or an angle, based on what we’re given, we can probably find it!

Let’s take a look at what Sine, Cosine, and Tangent are all about!

How to use SOH CAH TOA
Screen Shot 2020-07-04 at 5.04.02 PM

Now let’s see how we can apply trig ratios when there is a missing side or angle in a right triangle!

Screen Shot 2020-07-04 at 5.17.47 PM
How to use SOH CAH TOA
Screen Shot 2020-07-04 at 5.15.19 PM
How to use SOH CAH TOA
Screen Shot 2020-07-04 at 5.05.01 PM.png
Screen Shot 2020-07-04 at 5.19.13 PM
How to use SOH CAH TOA

Now for another type of question; using trig functions to find missing angles, let’s take a look:

Screen Shot 2020-07-04 at 5.19.51 PM
How to use SOH CAH TOA

Try the following Practice Questions on your own!

How to use SOH CAH TOA

Solutions:

Screen Shot 2020-07-04 at 5.06.37 PM.png

Still got questions?  No problem! Check out the video the same examples outlined above and happy calculating! 🙂

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Need to brush up on special right triangles? Check out this posts on the 45 45 90 special triangles here!

Perpendicular & Parallel Lines Through a Given Point: Geometry

Happy Wednesday math friends! Today we’re going to go over the difference between perpendicular and parallel lines, then we’ll use our knowledge of the equation of a line (y=mx+b) to see how to find perpendicular and parallel lines through a given point.  This is a common question that comes up on the NYS Geometry Regents and is something we should prepare for, so let’s go!

If you need any further explanation, don’t hesitate to check out the Youtube video below that goes into detail on how to find perpendicular and parallel lines through a given point one step at a time. Happy calculating! 🙂

Perpendicular Lines:

Perpendicular & Parallel Lines Through a Given Point

Perpendicular Lines: Lines that intersect to create a 90-degree angle and can look something like the graph below.  Their slopes are negative reciprocals of each other which means they are flipped and negated. See below for example!

Example: Find an equation of a line that passes through the point (1,3) and is perpendicular to line y=2x+1 .

Screen Shot 2020-06-10 at 10.28.20 AM
Perpendicular & Parallel Lines Through a Given Point
Perpendicular & Parallel Lines Through a Given Point
Screen Shot 2020-06-10 at 10.29.06 AM

Parallel Lines:

Parallel lines are lines that go in the same direction and have the same slope (but have different y-intercepts). Check out the example below!

Perpendicular & Parallel Lines Through a Given Point

Example: Find an equation of a line that goes through the point (-5,1) and is parallel to line y=4x+2.

Screen Shot 2020-06-10 at 10.34.46 AM
Screen Shot 2020-06-10 at 10.35.23 AM

Try the following practice questions on your own!

Practice Questions:

1) Find an equation of a line that passes through the point (2,5) and is perpendicular to line y=2x+1.

 2) Find an equation of a line that goes through the point (-2,4) and is perpendicular to lineScreen Shot 2020-06-10 at 11.24.06 AM

 3)  Find an equation of a line that goes through the point (1,6) and is parallel to line y=3x+2.

4)  Find an equation of a line that goes through the point (-2,-2)  and is parallel to line y=2x+1.

Solutions:

Screen Shot 2020-06-10 at 11.22.05 AM

Need more of an explanation? Check out the video that goes over these types of questions up on Youtube (video at top of post) and let me know if you have still any questions.

Happy Calculating! 🙂

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Looking for more on Perpendicular and parallel lines? Check out this Regents question on perpendicular lines here!

Median of a Trapezoid Theorem: Geometry

Hi everyone and welcome to Math Sux! In this post, we are going to look at how to use and applythe median of a trapezoid theorem. Thankfully, it is not a scary formula, and one we can easily master with a dose of algebra. The only hard part remaining, is remembering this thing! Take a look below to see a step by step tutorial on how to use the median of a trapezoid theorem and check out the practice questions at the end of this post to truly master the topic. Happy calculating! 🙂

*If you haven’t done so, check out the video that goes over this exact problem, also please don’t forget to subscribe!

Medians of a Trapezoid copy
Screen Shot 2020-06-02 at 7.31.07 AM

Step 1:  Let’s apply the Median of a Trapezoid Theorem to this question!  A little rusty?  No problem, check out the Theorem below.

Median of a Trapezoid Theorem

Median of a Trapezoid Theorem: The median of a trapezoid is equal to the sum of both bases.Step 2: Now that we found the value of x , we can plug it back into the equation for Screen Shot 2020-06-02 at 7.33.44 AMmedian,  to find the value of median Screen Shot 2020-06-02 at 7.34.25 AM

Screen Shot 2020-06-02 at 7.34.48 AM

Want more practice?  Your wish is my command! Check out the practice problems below:

Practice Questions:

Median of a Trapezoid Theorem
Median of a Trapezoid Theorem
Median of a Trapezoid Theorem

1.Screen Shot 2020-06-02 at 7.35.29 AMis the median of trapezoid ABCDEF, find the value of the median, given the following:2. Screen Shot 2020-06-02 at 9.01.08 AMis the median of trapezoid ACTIVE, find the value of the median, given the following:3.Screen Shot 2020-06-02 at 9.17.01 AMis the median of  trapezoid DRAGON, find the value of the median, given the following:

Median of a Trapezoid Theorem

4. Screen Shot 2020-06-02 at 9.23.08 AMis the median of trapezoid MATRIX, find the value of the median, given the following:

Solutions:

Screen Shot 2020-06-02 at 9.25.05 AM

Need more of an explanation?  Check out the detailed video and practice problems. Happy calculating! 🙂

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Area of a Sector: Geometry

Youtube Area of a sector copy

Hi math friends, has anyone been cooking more during quarantine?  We all know there is more time for cookin’ and eatin’ cakes but have you ever been curious about the exact amount of cake you are actually eating?! Well, you’re in luck because today we are going to go over how to find the area of a piece of cake, otherwise known as the Area of a Sector!

Now, we’ll all be able to calculate just how much we are overdoing it on that pie! Hopefully, everyone is eating better than I am (I should really calm down on the cupcakes).  Ok, now to our question:

*Also, If you haven’t done so, check out the video that goes over this exact problem, and don’t forget to subscribe!

Screen Shot 2020-05-19 at 4.18.42 PM

Explanation:

How do I answer this question? 

We must apply/adjust the formula for the area of a circle to find the area of the blue shaded region otherwise known as the sector of this circle.                                                    

How do we do this?    

Before we begin let’s review the formula for the area of a circle. Just a quick reminder of what each piece of the formula represents:

area of a sector

Step 1: Now let’s fill in our formula, we know the radius is 5, so let’s fill that in below:

Screen Shot 2020-05-19 at 4.26.08 PM

Step 2: Ok, great! But wait, this is for a sector; We need only a piece of the circle, not the whole thing.  In other words, we need a fraction of the circle. How can we represent the area of the shaded region as a fraction?

Well, we can use the given central angle value, Screen Shot 2020-05-19 at 4.27.17 PM, and place it over the whole value of the circle,Screen Shot 2020-05-21 at 4.01.12 PM . Then multiply that by the area of the entire circle. This will give us the value we are looking for!

area of a sector
area of a sector

Step 3: Multiply and solve!

Ready for more? Try solving these next few examples on your own to truly master area of a sector!

Practice Questions:

Find the area of each shaded region given the central angle and radius for each circle:

area of a sector

Solutions:

Screen Shot 2020-05-19 at 4.30.36 PM

What do you think of finding the area of sector? Are you going to measure the area of your next slice of pizza?  Do you have any recipes to recommend?  Let me know in the comments and happy calculating! 🙂

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Looking for more about circles? Check out this post on the circle formula here!

Math Resources (in the time of COVID)

Calling all students, teachers, and parents!  As everyone is stuck at home during a global pandemic, now is a great time we are all forced to try and understand math (and our sanity level) a little bit more.  Well, I may not be able to help you with the keeping sanity stuff, but as far as math goes, hopefully, the below math resources offer some much needed mathematic support.

All jokes aside I hope everyone is staying safe and successfully social distancing.  Stay well, math friends! 🙂

Kahn Academy: The same Kahn Academy we know and love still has amazing videos and tutorials as usual, but now they also have a live “homeroom” chat on Facebook LIVE every day at 12:00pm. The chats occur daily with Kahn Academy founder Sal and at times feature famous guests such as Bill Gates. Click the link below for more:

Math Resources

Khan Academy Homeroom 

Study.com: In a time when companies are being more generous, Study.com is here for us as they offer up to 1000 licenses for school districts and free lessons for teachers, students, and parents.  Check out all the education freebies here:

Study.com

Math Resources

Math PlanetIf you’re looking for free math resources in Pre-Algebra, Algebra, Algebra 2, and Geometry then you will find the answers you need at Math Planet.  All free all the time, find their website here:

MathPlanet 

Math Resources

MathSux: Clearly, I had to mention MathSux, the very site you are on right now! Check out free math videos, lesson, practice questions, and more for understanding math any way that works for you!

MathSux

What is your favorite educational site?  Let me know in the comments, and stay well! 🙂

Volume of a Cone: Geometry

The Voluminous “Vessel” at Hudson Yards

Calling all NYC dwellers! Have you seen the new structure at Hudson Yards?  A staircase to nowhere, this bee-hive like structure is for the true adventurists at heart; Clearly, I had to check it out!

Where does math come in here you say?  Well, during my exploration, I had to wonder (as am sure most people do) what is the volume of this almost cone-like structure? It seemed like the best way to estimate the volume here, was to use the formula for the volume of a cone!

What do you think the Volume is?

Volume of a Cone

Volume of a Cone:

I estimated the volume by using the formula of a three-dimensional cone. (Not an exact measurement of the Vessel, but close enough!) .

Volume of a Cone
Volume of a Cone

We can find the radius and height based on the given information above.  Everything we need for our formula is right here!

Volume of a Cone

Now that we have our information, let’s fill in our formula and calculate! 

Extra Tip! Notice that we labeled the solution with feet cubed Screen Shot 2019-04-14 at 4.53.49 PM.png, which is the short-handed way to write “feet cubed.”  Why feet cubed instead of feet squared? Or just plain old feet? When we use our formula we are multiplying three numbers all measured in feet:

radius X radius X (Height/3)

All three values are measured in feet! –> Feet cubed (Screen Shot 2019-04-14 at 4.53.49 PM.png)

  ____________________________________________________________________________________

Did you get the same answer? Did you use a different method or have any questions?  Let me know in the comments and happy calculating! 🙂

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Looking to apply more math to the real world? Check out this post on the Golden Ratio here!

Intersecting Secants: Geometry

intersecting secants theorem

Hey math friends! In today’s post, we are going to go over the Intersecting Secants Theorem, specifically using it to find the piece of a missing length on a secant line. We are also going to see proof as to why this theorem works in the first place!

Just a warning: this blog post contains circles. This usually non-threatening shape can get intimidating when secants, chords, and tangents are involved. Luckily, this question is not too complicated and was also spotted on the NYS Regents. Before looking at the questions below, here is a review on different parts of a circle. Pay close attention to what a secant is, which is what we’ll be focusing on today:

Screen Shot 2018-12-25 at 12.32.38 PM

Think you are ready? Let’s look at that next question!

Screen Shot 2018-12-25 at 12.34.09 PM
Intersecting Secants

What information do we already have? Based on the question we know:

Intersecting Secants
Intersecting Secants
Screen Shot 2018-12-25 at 12.54.24 PM.png

*Extra Tip! Why does this formula work in the first place!??  If we draw lines creating and proving triangle RTQ and triangle RPS are similar by AA, this leads us to know that the two triangles have proportionate sides and can follow our formula!         ___________________________________________________________________________________

Still got questions?  Let me know in the comments and remember having questions is a  good thing!

If you’re looking for more on intersecting secants, check out this post here for practice questions and more!

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The Magic of the “Golden Ratio”

Walking around NYC, I was on a mission to connect mathematics to the real world.  This, of course, led me to go on a mathematical scavenger hunt in search of  the “Golden Ratio.” Hidden in plain sight, this often times naturally occurring ratio is seen everywhere from historic and modern architecture to nature itself. 

What is this all-encompassing “Golden Ratio” you may ask?
It’s a proportion, related to a never-ending sequence of numbers called the Fibonacci sequence, and is considered to be the most pleasing ratio to the human eye.  The ratio itself is an irrational number equal to 1.618……..(etc.).

Why should you care?
When the same ratio is seen in the Parthenon, the Taj Mahal, the Mona Lisa and on the shores of a beach in a seashell, you know it must be something special!
The Golden Ratio The Golden Ratio The Golden Ratio The Golden Ratio

Random as it may seem, this proportion stems from the following sequence of numbers, known as the Fibonacci sequence:

1, 1, 2, 3, 5, 8, 13, 21, …….

Do you notice what pattern these numbers form?
(Answer: Each previous two numbers are added together to find the next number.)

Capture

The Golden RectangleThe most common example of the “Golden Ratio” can be seen in the Golden rectangle.  The lengths of this rectangle are in the proportion from 1: 1.618 following the golden ratio. Behold the beauty of the Golden Rectangle:

The Golden Ratio

How is the Fibonacci Sequence related to the Golden Ratio?                                               What if we drew a golden rectangle within our rectangle?

The Golden Ratio

Then drew another golden rectangle within that golden rectangle?

The Golden Ratio

And we kept doing this until we could no longer see what we were doing…….

The Golden Ratio
The Golden Ratio

The proportion between the width and height of these rectangles is 1.618 and can also be shown as the proportion between any two numbers in the Fibonacci sequence as the sequence approaches infinity. Notice that the area of each rectangle in the Fibonacci sequence is represented below in increasing size:

Where exactly can you find this Golden Ratio in real life? Found in NYC! The Golden ratio was seen here at the United Nations Secretariat building in the form of a golden rectangle(s).  Check it out!

Golden Ratio

Where have you seen this proportion of magical magnitude?  Look for it in your own city or town and let me know what you find! Happy Golden Ratio hunting! 🙂

If you’re interested in learning more about the golden ratio and are also a big Disney fan, I highly recommend you check out Donald Duck’s Math Magic!

Don’t forget to connect with MathSux on these great sites!

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Looking to learn more about math phenomenons found in the real world? Check out this article on fractals!