# Let’s Start Some Trig!

Happy New Year!

Usually at the beginning of the year, the PreCalc classes will start you on Trigonometry, which a lot of students loathe….It’s usually not too bad, but you may find that you have to memorize a little more for your trig sections. That’s OK!  Don’t be afraid to do some memorization in your math classes.

Did you know that the word “Trigonometry” means “Triangle Measure”?  That’s what trig is all about – measuring triangles (specifically, angles and sides of triangles).

I find that if the kids I tutor really liked Geometry, they usually prefer Trig to Algebra, since a lot of trig is visual.  But that doesn’t mean there isn’t a lot of algebra in trig…

Here are the sections I’ve completed on SheLovesMath.com on Trig; hope they help (and let me know if I’m missing anything).  Enjoy!   And check out my new Calculus sections – almost done!

Happy Math’ing,  Lisa 🙂

# Welcome Back to School – 2016!

School again!  How can that be?

Remember one thing: the teacher (and boss!) is always right (well, not always, but you know what I mean 😉 )  So hope you get a good math teacher, but it’s probably more important that you respect your teacher and always try to make a good impression (and work hard!).

I’ve posted these hints before, but they are worth mentioning again:

• Keep up with the homework; this is probably the most important.  Sometimes we think we understand the math, but actually we can’t do the problems.  Even if you have to do 2 or 3 from each section, make sure you can do each type of problem that is assigned to you.
• Try to go to as many school tutoring sessions that you can, if you don’t feel good about the math (or, even if you do!).  If you have the luxury of a private tutor, try to go ahead a little bit if you can, so you can hear the lecture twice.
• See if you can get your hands on old tests, like from an older sibling, friends, etc.  I know this may seem like “cheating”, but they do (or should) change the questions every year.  This way you can have a good idea what’s expected of you.
• Use She Loves Math or another site to read through material before you have the lecture; this way you’ll have a better chance of keeping up with the new material.   And feel free to send specific questions to me to see if I can help 🙂.

Hope this helps and let me know if you have any math questions; I’ll try to answer them!

I’m still not done with SheLovesMath, but getting there.  I’m still tackling Calculus sections, and I’ve done most of Differential Calculus and started Integral Calculus.  The Calc pages completed are Definition of the Derivative, Basic Differentiation Rules, Equation of the Tangent Line, The Chain Rule, Implicit Differentiation and Related RatesCurve Sketching, Optimization, Differentials, Linear Approximation and Error Propagation, Antiderivatives and Indefinite Integration, U-Substitution Integration, and Differential Equations and Slope Fields.

Please spread the word about She Loves Math, and let me know how it can be better!  I have almost 2 million pages views total now, and I hear from students all over the world 😉 .

Lisa

# How to Prepare for Finals

Hey Math Geeks!

It’s been awhile since I’ve posted; sorry about that!  I have been plugging away and have completed more Pre-Calc sections (Sequences and Series, Binomial Expansion and Introduction to Limits), more Trig sections (Linear and Angular Speeds, Area of Sectors, and Length of Arcs), and have started Calculus!

The Calc pages completed are Definition of the Derivative, Basic Differentiation Rules, Equation of the Tangent Line, The Chain Rule, Implicit Differentiation and Related Rates, and starting Curve Sketching.

Here’s some tips I have for finals that I wrote about last year:

Trust me, I’ve haven’t aced every math test I’ve taken.  Both as a student (ions ago!) and a math tutor, I’ve found that these things really help when “studying” for taking a math test or final.  The main thing is that you don’t “study”, you “work problems” to prepare:

1. If you are lucky enough to have your old tests for the semester (or year, if the final covers the whole year), retake them again!!  Cover up the answers and go for it!  I know it seems like a lot of work, but it’s the surefire way of knowing if you really get the stuff.  If you don’t have time to go through all the tests, go through the ones that were at the beginning of the learning period and also the ones where you didn’t do the greatest.
2. If you can’t get your old tests back, try to work any packets, homework, or chapter reviews covering the material.  Just pick out one or two problems covering each concept.
3. If you’re having trouble with any of the concepts when working problems, create a “cheat sheet” (not for the test!  lol) of notes that you need to study.  Remember that some things (actually a lot of things) in math you have to memorize – that’s OK.
4. Glance over your notes one more time to see if you missed anything after doing steps 1 and 2 above.  This is not good enough though; you may think you know how to work a problem by looking at your notes, but you don’t really know unless you actually try it!  If you’re unsure about a concept, you might even check out She Loves Math!
5. One more thing – Reviews can be deceiving!!  Don’t rely on them; many times, they don’t cover everything on the test!

Good luck, and let me know if you have any questions about anything.  And of course, when you’re taking the test, skip over any problems that stump you, put a big circle on it, and come back to it.  Time can be your enemy with math tests and we’ve all had those brain lapses.  Try not to let those times affect the rest of the test.

Happy Mathing and have a wonderful summer!  I hope to do a lot of She Loves Math writing this summer!

Lisa 🙂

# Welcome Back to School!!

Well it’s here already – school again!   I always love the beginning of a new school year, in spite of summer being over….   I hope everyone ends up with a good math teacher 🙂

I’ve been busy writing more of She Loves Math; I never get as much done as I’d like, but here are some new sections I’ve added:   Trig sections, including Right Triangle Trig, Angles and the Unit Circle, Graphs of Trig Functions, Transformations of Trig Functions,  Inverses of Trig Functions, Solving Trig Equations, Trig Identities, Polar Coordinates, and Trig and the Complex Plane.   I’ve also completed  Conics, Systems of Non-Linear Equations, Introduction to Vectors, and Parametric Equations, and working on Sequences and Series.

I thought I’d give you a few hints and tips to have a productive and successful year in math:

• Keep up with the homework; this is probably the most important.  Sometimes we think we understand the math, but actually we can’t do the problems.  Even if you have to do 2 or 3 from each section, make sure you can do each type of problem that is assigned to you.
• Try to go to as many school tutoring sessions that you can, if you don’t feel good about the math (or, even if you do!).  If you have the luxury of a private tutor, try to go ahead a little bit if you can, so you can hear the lecture twice.
• See if you can get your hands on old tests, like from an older sibling, friends, etc.  I know this may seem like “cheating”, but they do (or should) change the questions every year.  This way you can have a good idea what’s expected of you.
• Use She Loves Math or another site to read through material before you have the lecture; this way you’ll have a better chance of keeping up with the new material.   And feel free to send specific questions to me to see if I can help 🙂

Hope this helps!

Please spread the word about She Loves Math, and let me know how it can be better!   And I’ll try to post blogs more often 🙂

Lisa

# Happy π Day!

Today, 3/14/15, at 9:26:53, we experience a once-in-a-lifetime event:  the date and time will be the first 10 digits of the number π!

I just saw on the news that there are some kids who can repeat the digits of π to over 2000 digits!   Much better than I can do (I’m lucky if I can get up to 10 digits 😉

She Loves Math is progressing, and now gets 3000-5000 hits a day.   I’ve completed The Inverse Trig Functions, Solving Trig Equations, Trig IdentitiesLaw of Sines and Cosines, and almost done with Polar Graphs.    Earlier trig sections includes Right Triangle Trig, Angles and the Unit Circle, Graphs of Trig Functions, and Transformations of Trig Functions.

Have some pie today!

Lisa

# Converting Quadratic Functions, and Preparing for Math Finals

First of all, I’ve been fortunate enough to be able to share a valuable article from Nghi H. Nguyen, who is a reader of my site.  The article covers “the new Transforming Method for solving quadratic equations” and is quite interesting:

Convert quadratic functions from one form to another, by Nghi H. Nguyen

Here is one more article supplied by Nghi:

Solving Quadratic Equations by the New Transforming Method, by Nghi H. Nguyen

Also, looks like another semester is about to come to a close and I’d like to repeat my math test-taking hints that talked about in an earlier blog:

Trust me, I’ve haven’t aced every math test I’ve taken.  Both as a student (ions ago!) and a math tutor, I’ve found that these things really help when “studying” for taking a math test or final.  The main thing is that you don’t “study”, you “work problems” to prepare:

1. If you are lucky enough to have your old tests for the semester (or year, if the final covers the whole year), retake them again!!  Cover up the answers and go for it!  I know it seems like a lot of work, but it’s the surefire way of knowing if you really get the stuff.  If you don’t have time to go through all the tests, go through the ones that were at the beginning of the learning period and also the ones where you didn’t do the greatest.
2. If you can’t get your old tests back, try to work any packets, homework, or chapter reviews covering the material.  Just pick out one or two problems covering each concept.
3. If you’re having trouble with any of the concepts when working problems, create a “cheat sheet” (not for the test!  lol) of notes that you need to study.  Remember that some things (actually a lot of things) in math you have to memorize – that’s OK.
4. Glance over your notes one more time to see if you missed anything after doing steps 1 and 2 above.  This is not good enough though; you may think you know how to work a problem by looking at your notes, but you don’t really know unless you actually try it!  If you’re unsure about a concept, you might even check out She Loves Math!
5. One more thing – Reviews can be deceiving!!  Don’t rely on them; many times, they don’t cover everything on the test!

Good luck, and let me know if you have any questions about anything.  And of course, when you’re taking the test, skip over any problems that stump you, put a big circle on it, and come back to it.  Time can be your enemy with math tests and we’ve all had those brain lapses.  Try not to let those times affect the rest of the test.

Happy Mathing,

Lisa

# Welcome to Trig!

Welcome from She Loves Math!   I know it’s been awhile, but I wanted to welcome everyone back to school and talk a little about Trigonometry, which is usually part of a Pre-Calc class these days.

I’ve completed the Right Triangle Trigonometry, Graphs of Trig Functions, Transformations of Trig Functions sections, and the beginning of the Inverses of the Trigonometric Functions section.

Here is an excerpt from my Right Triangle Trigonometry section to get you started with Trig:

You may have been introduced to Trigonometry in Geometry, when you had to find either a side length or angle measurement of a triangle.  Trigonometry is basically the study of triangles, and was first used to help in the computations of astronomy.  Today it is used in engineering, architecture, medicine, physics, among other disciplines.

The 6 basic trigonometric functions that you’ll be working with are sine, cosine, tangent, cosecant, secant, andcotangent.  (Don’t let the fancy names scare you; they really aren’t that bad).

With Right Triangle Trigonometry, we use the trig functions on angles, and get a number back that we can use to get a side measurement, as an example.    Sometimes we have to work backwards to get the angle measurement back so we have to use what a call an inverse trig function But basically remember that we need the trig functions so we can figure out triangles’ sides and angles that we don’t  otherwise know.

Later, we’ll see how to use trig to find areas of triangles, too, among other things.

You may  have been taught SOH – CAH – TOA  (SOHCAHTOA)  to remember these.  Back in the old days when I was in high school, we didn’t have SOHCAHTOA, nor did we have fancy calculators to get the values; we  had to look up trigonometric values in tables.

Remember that the definitions below assume that the triangles are right triangles, meaning that they all have one right angle.  Also note that in the following examples, our angle measurements are in degrees; later we’ll learn about another angle measurement unit, radians, which we’ll discuss here in the Angles and Unit Circle section.

## Basic Trigonometric Functions (SOH – CAH – TOA)

Here are the 6 trigonometric functions, shown with both the SOHCAHTOA and Coordinate System Methods.   Note that the second set of three trig functions are just the reciprocals of the first three; this makes it a little easier!

Note that the cosecant (csc), secant (sec), and cotangent (cot) functions are called reciprocal functions, or reciprocal trig functions, since they are the reciprocals of sin, cos, and tan, respectively.

For the coordinate system method, assume that the vertex of the angle in the triangle is at the origin (0, 0):

Keep checking back, as I’m trying to write every day, and please let me know if you have any questions or comments.    Keep working hard!   Lisa 🙂

# Conics Section Done – Starting Trigonometry Sections

Sorry that it’s been awhile, guys 😉 .

Anyway, just finished the Conics: Circle, Parabolas, Ellipses, and Hyperbolas section; it was a long one.

Also, been working on some Trig sections: finished Right Triangle Trigonometry, and starting Angles and the Unit Circle – this will be an important one!!  Hope to finish that one by the start of the new school year!

You may notice that I’ve been playing around with Adsense on the site.  Hmm….thought I wouldn’t do this, but it’s nice to have a little money coming on, at least to pay for my expenses to keep the site up and running…..

As always, I love to hear from you about what you like and don’t like about the site, so keep those comments coming…..

Happy Math’ing,

Lisa

# Math Tips: Exponents and Logs

Sorry it’s been awhile since I’ve posted my math tips – it’s always a little crazy getting through the holidays.  I can’t believe in a couple of months we’ll be finished with yet another school year 😉

She Loves Math continues to attract new viewers; I’m getting 1500-2000 hits a day now.  I’ve loved hearing from you and trying to help you solve math problems.

I just completed the Exponential and Logarithmic Functions section; it took me a little longer than expected – sorry about that.  Here is a short article I wrote on exponents and logs for someone; hope it’s useful:

## What Is an Exponent?

First, we first need to understand what an exponent is.  An exponent just means that a number is multiplied that many times by itself.

Let’s use the following example.   If your savings account gives you interest at 5% every year (a rarity, these days!), every year your savings would be multiplied by 1.05.  This is because you get to keep what you originally put in your account, your principle (let’s say $100), and add 5% (or .05) times your amount every year. After the first year, your account would have 100(1.05) or$105.  After the second year, your account would have 100(1.05)(1.05) or  which is $110.25. Do you see how, in the nth year, your account would have dollars? (“n” is the exponent in this case.) It’s a confusing concept, but if you do the math year-by-year, it makes sense. By the way, this concept of multiplying your principle by an interest amount every year is called compounding. (Start saving and investing your money early in life; it pays!) ## How Do Exponents Relate to Logs? Now, let’s get back to logs. Logs are related to exponents; in fact, the log of a number is an exponent. Look at the following graphic to see how exponents and logs are related. Remember that every exponent and log has a base, as designated by “b” in this diagram. Our base in the example above is 1.05. Note: If there is no “b” next to the log, then the base is assumed to be 10. ## Why Do We Need Logs?: One Example We are going to show an example of using logs in algebra (the branch of mathematics where letters represent values). But why would we ever want to change an exponential algebraic expression or equation to a logarithmic one? One important purpose of logs is for solving mathematical equations with unknowns or variables in an exponent. Let’s say we had a beginning amount in our back account ($100) and had an ending account (say, $250), and we wanted to know how many years it took to reach that amount? We could use the “guess and check” method to try different numbers in the exponent to get the$250 amount.  But the easiest way to get the number of years, or the exponent in the equation  is to use logs.  So a useful application of using logs is to solve for variables, or unknowns, that are in exponents.

There are certain properties of logs that very helpful in solving equations.  One of the most useful properties is that the log of a number raised to an exponent is equal to that exponent times the log of the number without the exponent:  .  This is called the Power Property or Power Rule of logs.  This is an important property since it helps “get the exponent down” so we can solve for it using algebra.

## Solving a Logarithmic Equation Using Algebra

Let’s go ahead and use logs, including the Power Property, to solve the algebraic equation  to get the number of years that it would take to go from $100 to$250 with a yearly interest rate of 5%.  We want A (the ending amount) equal to $250: So the number of years it would take to have$250 in your account is 18.78 years.  This method is much easier than the “guess and check” method that involves just trying different numbers in the exponents and seeing if you get the \$250.  And we can check our answer by using our calculator:  .

And this is just one of many applications of using logs; they are also used in biology (population growth), anthropology (carbon dating), geology (the Richter scale for earthquakes), chemistry (pH scales) and many other fields.

## So That’s Why We Need Logs!

Logs are here to stay and are extremely powerful in mathematics. They are used in everyday applications of science and finance; in fact, logs are used at your local bank to help with the interest in your bank.  And again you can refer to the Exponential and Logarithmic Functions section of She Loves Math to learn more!

# Math Tip of the Week – Multiplying and Factoring Polynomials

A couple of techniques that you’ll have to learn early in your high school math career is multiplying and factoring polynomials.  Polynomials are a collection of coefficients and variables with exponents added and subtracted together, but none of the variables appear in denominators.

You may have heard of the “foiling” process, which they actually don’t teach as much anymore.  Nevertheless, I still feel it’s very important to learn, since it’s then easier to “unfoil” (factor):

### FOILING

FOIL stands for First Outer Inner Last.  Just remember that we multiply the First Outer Inner and then Last terms and put plus signs between them (unless the product is negative).  Here are some examples.   Note that FOILing only works if you multiply binomials – each factor has two terms.

Note the last two examples are “special cases” that you’ll see a lot: difference of two squares, and perfect square trinomials; note the shortcuts with these cases.

### “PUSHING THROUGH” OR DISTRIBUTING TERMS OF POLYNOMIALS

This technique seems more popular these days, since it can apply to polynomials of any size.  Really, what we are doing when we are FOILing is using the distributive method to make sure every term (variable or number) is multiplied by every other one and then you add them all up.  We can also think of this as “pushing through” the terms to every other term.

Also called “double distributing”, this way of multiplying binomials is more popular now, since it can be used for any polynomial.

Let’s do a couple of the problems above and also see how “pushing through” can be used with polynomial products when we don’t have two binomials:

More multiplying polynomials and also factoring polynomials, including by Completing the Square, can be found in the Introduction to Multiplying Polynomials,  Introduction to Quadratics and the Factoring Quadratics and Completing the Square sections.

Enjoy!!