Howdy math friends! In this post, we are going to learn about solving trigonometric equations algebraically. This will combine our knowledge of algebra and trigonometry into one beautiful question! For more on trigonometric functions and right triangle trigonometry check out this post here.
Solving trigonometric equations. Sound complicated? Well, you are correct, that does sound complicated. Is it complicated? Hopefully, you won’t fund it that way after you’ve seen this example. We are going to do this step by step in the following regents question:
How do I answer this question?
The questions want us to solve for x.
Step 1: Pretend this was any other equation and you wanted to solve for x. Thinking that way, we can move radical 2 to the other side.
Step 3: Now we need a value for x. This is where I turn to my handy dandy reference triangle. (This is a complete reference tool that you should memorize know). For more on special triangles, check out this post here.
Step 5: Notice that cosine is positive in Quadrants I and IV. That means there are two values that x can be (one in each quadrant). We already have x=45º from Quadrant I. In order to get that other value in Quadrant IV, we must subtract 360º-45º=315º giving us our other value.
Does this make sense? Great! 🙂 Is it clear as mud? I have failed. But I have not given up (and neither should you). Ask more questions, look for the spots where you got lost, do more research and never give up! 🙂
Hopefully you enjoyed my short motivational speech. For more encouraging words and math, check out MathSux on the following websites! Sign up for FREE math videos, lessons, practice questions, and more. Thanks for stopping by and happy calculating! 🙂
Greetings math friends, students, and teachers I come in peace to review this piecewise functions NYS Regents question. Are they pieces of functions? Yes. Are they wise? Ah, yeah sure, why not? Let’s check out this piecewise functions NYS Regents question below and happy calculating! 🙂
1 value satisfies the equation because there is only one point on the graph where f(x) and g(x) meet.
Does the above madness make sense to you? Great!
Need more of an explanation? Keep going! There is a way to understand the above mess.
How do I read this?
Looking at this piece-wise function: We want to graph the function 2x+1 but only when the x-values are less than or equal to negative 1.
We also want to graph 2-x^2 but only when x is greater than the negative one.
One way to organize graphing each piece of a piecewise function is by making a chart.
Lets start by making a chart for the first part of our function 2x+1:
Is it all coming back to you now? Need more practice on piece-wise functions?Check out this link here and happy calculating! 🙂
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Hey math friends! In this post, we are going to go over Factor by Grouping, one of the many methods for factoring a quadratic equation. There are so many methods to factor quadratic equations, but this is a great choice, for when a is greater than 1. Also, if you need to review different types of factoring methods, just check out this link here. Stay curious and happy calculating! 🙂
Before we go any further, let’s just take a quick look at what a quadratic equation is:
Usually, we can just find the products, the sum, re-write the equation, solve for x, and be on our merry way. But if you notice, there is something special about the question below. The coefficient “a” is greater than 1/. This is where factor by grouping comes in handy!
Now that we know why and when we need to factor by grouping lets take a look at our Example:
Factor By Grouping: Hard to solve? No. Hard to remember? It can be, just remember to practice, practice practice! Also, if you are in need of a review of other methods of factoring quadratic equations, click this link here.
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. Happy Calculating! 🙂
Ahoy math friends and welcome to Math Sux! In this post, we are going to learn about the equation of a circle, mainly how to write it when given a circle on a coordinate plane. We’ll see how to find the radius and center from the graphed circle and then see how to transfer that into our equation of a circle. And we’re going to do all of this step by step with the following Regents question. Stay curious and happy calculating! 🙂
How do I Answer this Question?
Let’s mark up this coordinate plane and take as much information away from it as possible. When looking at this coordinate plane, we can find the value of the circle’s center and the circle’s radius.
Now that found the center and radius, we can fit each into our equation:
The only equation that matches our center and radius, is choice (2).
If the above answer makes sense to you, that’s great! If you need a little further explanation, keep going!
Equation of a Circle:
Equation of a Circle: Let’s take a look at our answer and break down what each part means:
So what do you think of circles now? Not to shabby, ehh? 🙂
Looking for more on circles? Check out this post on how to find the Area of a Sector here!
Howdy math peeps! In this post, we are going to go over the recursive formula step by step by reviewing an old Regent’s question. These things may look weird, confusing, and like a “what am I doing?” moment, but trust me they are not so bad! We are going to take a look at the Regents question below and then find the correct recursive formula that follows the given sequence by going through each answer choice. Before we begin to answer our questions though, we will first define and break down what a recursive sequence is. Also, be sure to check out the video at the end of this post for more examples and further explanation. Happy calculating!
Before we dive into the solution to this question, let’s first look at a recursive formula example and define what they are in the first place!
What is the Recursive Formula?
A Recursive Formula is a formula that forms a sequence based on the previous term value. All this means is that it uses a formula to form a sequence-based pattern. Recursive formulas can take the shape of different types of sequences, including arithmetic sequences (sequence based on adding/subtracting numbers) or geometric sequences (sequence based on multiplying/dividing numbers). Check out the example below:
a1=2 , an+1=an+4
How do You Solve a Recursive Equation?
When solving a recursive equation, we are always given the first term and a formula. We start by using the first given term of our sequence, usually represented as a1, and plug it into the given formula to find the value of the second term of the sequence. Then we take the value of the previous term (the second term in our sequence) and plug it into our formula again, to find the third value of our sequence…. and the pattern continues! The solutions we get from each step up forms a sequence. If this sounds confusing, don’t worry because we are going to look at an example! Take a look at the recursive formula below:
a1=2 , an+1=an+4
Now let’s take another look at our recursive formula, this time breaking down what each part means:
a1 always represents the first given term, which in this case is a1=2. Next, we plug in 2 for an into our formula an+4 to get (2)+4=6. This gives us our second term in the sequence, which is 6. Next, we plug in 6 into the formula to get (6)+4=10, which is the value of the third term in the sequence. And we continue the pattern, always taking the value of the preceding term! In this case, I just found the first few terms below, but the recursive formula can continue infinitely! Again , if this sounds confusing, please take look at the pattern below:
Terms of the Sequence: 2, 6, 10, 14, 18….
Notice all the terms of the sequence are circled in pink, forming a sequence of, 2, 6, 10, 14, 18!? This is what the recursive formula produces!
Now back to our Original Question:
Q: What recursively-defined function represents the sequence 3, 7, 15, 31,
(1) f(1) = 3, f(n + 1) = 28 (n) +3
(2) f(1) = 3, f (m + 1) = 28(0) – 1
(3) f(1) = 3, f(n + 1) = 28 (n) +1
(4) f(1) = 3, f (n + 1) = 38 (n) -2
How do I answer this question?
At first glance, all of these answer choices may look exactly the same as there are many recursive formulas to choose from. The first thing we need to do is to identify how each answer choice is different. Notice below, the section highlighted in green? This is what we will focus on for finding the correct recursive formula!
The question we are working with actually gives us a sequence and we need to find the recursive formula that works with it! Our goal with this question is to work backward to test out each recursive formula given to us until we get the correct sequence. To begin, let’s first identify each term in our given sequence.
As we go through each answer choice, we are looking for the recursive formula that gives us the above sequence 3, 7, 15, 31. Let’s start with choice (1) which happens to be a type of geometric sequence.
Right away we can see the sequence forming for choice (1) is 3, 11, … where the first term, 3, matches our original sequence, but the second term we get which is, 11, does not. This means we will need to move on and find the sequence of the next option, choice (2), which happens to be another type of geometric sequence. Let’s take a look:
For choice (2), we can see that the sequence we get is 3, 7, 127 which matches our given sequence for the first two terms 3, 7, but does not match the third term 127, when we need a 15 here (the original sequence is 3,7,15,31). Thus, we must move onward to the next recursive formula by testing out choice (3), which this time is an arithmetic sequence! Maybe we will have better luck!
This last option, choice (3) provided us with the same sequence we were originally provided within the original question, 3, 7, 15, 31. We have found our answer and now we can celebrate!
Recursive Formula Examples
Still have questions about recursive formulas? Check out more on recursive formula examples here and in the video above! And if you’re looking to learn all there is to know about sequences, check out this post here! Looking to move ahead? Check out the infinite geometric series lesson here! Also, please don’t hesitate to comment with any questions. Happy calculating! 🙂