Weekly Math Problem! #24

Trigonometric Equation. The last time I had a trigonometric equation on here was back in June, WMP! #8. (Be sure to check it out. 😉)  That trig equation didn't have any of the functions squared. This week's problems involves a squared function and a function with a double angle. I know, I know... Why, Lori?! Because it looked fun to solve. 🤓

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️ Double angle formulas
        ✔️ How to solve quadratic equations
        ✔️ How to isolate the angle in a trigonometric function

By the way, you'll definitely need at least a scientific calculator for this one. Here is WMP! #24:

Happy solving!

Check back on Friday, October 23rd for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

⬇️

⬇️

⬇️

⬇️

⬇️

This problem is a quadratic trigonometric equation. What makes it "difficult" to solve is the trigonometric function with the double angle. How do we deal with that? Use a double angle formula to replace it in the equation. Let me show you...

                                                                       

As you can see, switching out the double angle trigonometric function for an equivalent one helped a great deal. It allowed for the original equation to be reduced to a simpler quadratic equation that could be solved with ease...and a calculator...lol 😅.

 

▪️ Did you use the same double angle formula I used to solve?? 

▪️ Comment below with your responses and let me know what you thought about this week's problem.

Thanks for solving with me this week!

See you next week for WMP! #25

👍🏿

Cheers!

The Younge Lady

Weekly Math Problem! #19

Oblique Triangles. Back to some good ol' trig. 👍🏿 Different than special triangles, oblique triangles don't really have ways to come up with any of the values for the angles or legs of the triangles by following "shortcuts". You pretty much have to rely on the Law of Sines or Law of Cosines to get the answers to the problems. 

To solve this week's problem in completion, you need to recall the following math skills:

    ✔️  When/How to use the Law of Sines

    ✔️  When/How to use the Law of Cosines

    ✔️  How to find the area of an oblique triangle

    

By the way, you'll need a scientific calculator to help you in computing.

WMP! #19 says...

Happy solving! 

Check back on Friday, September 11th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

⬇️

⬇️

⬇️

⬇️

⬇️

Where to start? Well, there is not enough information to solve for any of the unknown angles, B or C. So, we'll solve for the missing side, a. One thing to notice is that the given triangle is not a right triangle, therefore, we cannot use the Pythagorean theorem. That's okay. We'll use something that looks a lot like it--the Law of Cosines formula. We'll use the one that fits our given information.

Now that all the sides are known, we have enough information to solve for the remaining angles, B and C. You have the option to use the Law of Cosines formula again, or you can use the Law of Sines. There's a catch...🤨. Sorry. 🤷🏿 . If you choose to use the Law of Sines, then you will run into a problem solving for angle B. Why?? "A triangle with two sides and a consecutive angle is the ambiguous case of the Law of Sines. In this case, you may have one, two, or no solutions, (Pre-Calculus Workbook for Dummies, Gilman, et al., 2009, p. 179)." Using the Law of Sines to solve for angle C is just fine. If you continue to solve for the remaining angles using the Law of Cosines, you also be fine. What did I do? I used the Law of Sines to solve for angle C.

Now that a second angle has been found, we can easily use subtraction to solve for the last angle:

It's done...the triangle has been solved for! Here's the triangle summary:

The last thing the needs to be done is to find the area of the triangle. Since the height of the triangle cannot be easily seen or found, we will not be using the traditional area formula of a triangle. Here's another formula to find the area of an oblique triangle:

The area can also be found by replacing the sides and angle, accordingly, in the formula. Similar results will be found.

 

◾️ How did you do in solving this week's problem?? 

◾️ Comment below with your responses and let me know what you thought about this week's problem.

Up next... WMP! #20❗️

Cheers!

The Younge Lady

Weekly Math Problem! #18

Fourier Series. A couple posts ago, WMP! #16 to be exact, I mentioned that I learned about Fourier Series over the summer. "Just like that?!," you may ask. Well, no. Actually, I was enrolled in a PDE (Partial Differential Equations) course where Fourier Series was one of the topics we learned...in 8 weeks. 😯  Let me tell you... 👏🏿those 👏🏿 eight 👏🏿 weeks 👏🏿 were 👏🏿 no 👏🏿 joke! [Background: The last time I was in an ODE (Ordinary ...) course, I audited it about about 8/9 years ago.)] Definitely, I was reminded of Taylor and Maclaurin series when we got to Fourier, so that's why I did a Taylor series question recently. I felt like I needed to go over it and to make sure that last time I saw it was weeks ago and not years ago. 

I find that the calculation aspect isn't too bad, especially if you have a good integral calculus background. What can be tricky, as the professor I had reminded us, is knowing the theory well enough to move forward in solving a question. 

To solve this week's problem in completion, you need to recall the following math skills:

    ✔️  How to deal with piecewise functions

    ✔️  Integration techniques

    ✔️  How to evaluate trigonometric functions

    

WMP! #18 asks to...

Happy solving! 

Check back on Friday, September 4th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

⬇️

⬇️

⬇️

⬇️

⬇️

Are you ready for this?? I hope so, because I'm still not ready even though I have it done. 😅 Here goes...

Here is graph of the function over the interval [-3π, 3π]:

**This plot was created using GeoGebra's Geometry App.

To begin coming up with the Fourier Series for this function, these are the components that are needed:

The first calculation consisted of finding the coefficient ak in two situations--when k=0 and when k≠0. As you see below, both situations yielded zero.

Next up, the calculation for the coefficient bk. Ahhh...this is where things get interesting. Why? Oh....just take a look 👀... 

The calculation for bk is alternating! An even k does nothing for the series, but an odd k does! To see what the Fourier series, S(x), is for f(x), I put in the various calculation results then simplified.

As you can see, cosine disappears from the final series, leaving sine. I switched out the expression for k with an expression for m that generates odd natural numbers.

Below are four series corresponding to the values where m ends at 1, 2, 3, and 7, respectively. Consequently, m matches how many terms each series has. (I hope that makes sense.) 

I graphed the four series along with the original function, so that it can be seen how each series looks in relation to the original unction. The more terms the series has, the more it closely resembles the original function.

**This plot was created using GeoGebra's Geometry App.

Whew...this was a long one, but a good one. 

◾️ Have you ever learned Fourier series or any or topics from PDE?? 

◾️ Comment below with your responses and let me know what you thought about this week's problem.

WMP! #19 is next❗️

Cheers!

The Younge Lady

Weekly Math Problem! #3

Verifying Trigonometric Identities. This week's problem comes from an Instagram post...imagine that! 😀  I was scrolling through my feed, like I normally do, and saw @mathematicalmodels's post. Of course I was intrigued. I saw a math problem with trig and said to myself, "Oooo...what's this?" She re-posted from Canadian teen star Maitreyi Ramakrishnan who shared this old test question.

I like these kinds of trig problems. They're like puzzles to me. To solve this week's problem in completion, you need to recall the following math skills:

     ✔️  How to use trigonometric formulas/identities 
     ✔️  Adding/subtracting fractions 

Here goes WMP #3! 

Happy proving! Check back on Friday, May 22nd for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

⬇️

⬇️

⬇️

⬇️

⬇️

There is no "one way" to prove/verify this problem. In @mathematicalmodels's post, the solution was completed using the angle-addition formulas. I decided to go with the product-to-sum angle formulas just to see how it would look. Take a look for yourself...

◾️ Would you have done the problem differently?? 

◾️ How do feel about trigonometry...love it, hate it, or are you somewhere in the middle??

◾️ Comment below with your responses and let me know what you thought about this week's problem.

See you soon for WMP #4! 😉😁

Cheers!

The Younge Lady

Weekly Math Problem! #1

It's SPRING! It's been spring for over a month now, so it's about time I get these problems going. (I know..I know...it's been a long wait. However, I have an excuse. It's the same as everybody else....and you know what it is. 👀) For a little more info on these problems, check out my previous post. As promised, The Younge Lady's weekly math problems are here! So, without further ado, here is WMP #1:

Limits Involving Trigonometric Functions. This week's question came from a tutoring session that happened a couple months ago and is typically learned in a Calculus I course. To solve it in completion, you need to recall the following math skills:

     ✔️  How to deal with fractions 
     ✔️  How to deal with trigonometric functions 
     ✔️  Limit properties
     ✔️  Finding derivatives of trigonometric functions 

I will say that there are two ways to do this problem. Ready?

Happy solving! Check back on Friday, May 8th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

⬇️

⬇️

⬇️

⬇️

⬇️

As previously mentioned, there are two ways to find this limit. The first method is algebraic in nature; it is not necessary to know derivatives to find this limit. Limits are most often taught before derivatives are, so algebraic methods are explored. Once derivatives and other topics have been taught, limits are re-explored at some point and the method of L'Hôpital's rule is used to evaluate limits that yield indeterminate forms when evaluated at the specified value.

Algebraic Method

L'Hôpital's Rule Method

That's pretty much it. Two different methods, same answer. That's one of my favorite things about mathematics--one problem can be solved multiple ways. 

◾️ Did you arrive at the same answer I have above?? 

◾️ How did you find the limit?? 

◾️ When was the last time you did a limit problem??

◾️ Comment below with your responses and let me know what you thought about this week's problem.

Cheers!

The Younge Lady