Graphing Trig Functions

We worked mostly on problems with trigonometric functions.

A few involved graphing the functions. It can help quite a bit to know the shape of the graph in advance. But, if you do not, or even if you somewhat do, then you can use


Also used some double angle formulas and saw how you can use the trig functions with sums for double angle formulas.

Infinite sums, getting towards the idea of integration

Worked on problems related to the early ideas of calculus today.

The ideas today were closer to integration, finding an area using a summation of infinitely small rectangles using limits.

That meant finding the area of one rectangle and adding up all of them. So a base multiplied by a width.

The actual shape was a triangle and rectangle, but the technique becomes more useful for shapes which are not easy to calculate.

There were quite a few steps, so a single mistake could throw everything off.

RLS Friday sheet Precalculus tutoring – fourth order polynomial with complex numbers

We started by talking about the Lorentz Force Law which makes many problems with forces and magnetism make more sense in my opinion.

Then looked at a Friday sheet.

Did a problem with complex numbers starting with a fourth order polynomial  that had two zeros given. It involved either long division or synthetic division, we used long division.

Another problem had logarithms. e^x and lnx are functions that counteract each other.

The half angle formula was used for another problem.

Starting to look at limits in precalculus

We looked at limits in the context of the last few weeks of precalculus.

All of the functions we saw had a (single) removable discontinuity or hole. The limits were found for that point and two other points. For the limit to exist, it needs to go to the same place from the right and from the left.

It didn’t exactly make sense to find a limit for points that did not have holes, but that was part of some of the problems.

Tutoring Precal for RLS, Double angle formulas and graphing trig functions

We started by looking at homework which used the double angle formula.

The basic approach was to do two things, could start in either order.

1. Draw the angle with a right triangle on the xy axes.
2. Use a double angle formula which is convenient

You can figure out the ratios for the trig functions using

Then looked at a previous test with graphing trig functions. Should spend more time on that. But one thing you can do most of the time is plot points by plugging in x values and getting y values. You use again,


It does help quite a bit to know the basic shape of the functions.

Tutoring RLS Precalculus, trig functions and Friday sheet

We looked at some problems with trig functions, especially sum and difference formulas.

For many of them, the angle does not need to be known if the sides of the triangle are known.

Used the half angle formula a bit. Reviewed the quadrants and which trig functions are positive and negative in different positions.

Then got into the Friday sheet which dealt with logarithms a bit.

Tutoring Precalculus, Series and Sigma Notation

We mostly looked at problems with sigma notation. At the bottom of the capital (Greek) letter Sigma, there is a variable with the starting point and at the top a finishing point if it is finite or infinity.

Then the sequence of numbers goes into an equation that are added together.

For a geometric series, there can be a finite sum or it can diverge.

Exponents can be used, and series can also alternate between positive and negative. You can describe the same series in ways that look different.

Looked a bit at combinations and permutations, including one that required a somewhat seldom used formula at least in high school math classes.

“Is the square root of -1 equal to just 1?”

There’s a way to check that. Or any similar problem with a square root.

Take what you think is the answer, 1 and square it. If you get -1, then yes, it is. If not, it isn’t.

1(1) = 1

Not -1, so no it is not.

The solution is an imaginary number. Using both real numbers and imaginary numbers is called ‘complex’.

The square root of -1 is called i. So i squared is -1.

Matrix Addition, Subtraction, Multiplication – Tutoring Math

We worked on matrices.

Starting with matrix addition and subtraction, which were not a problem.

Then got into multiplication  by numbers as well as addition and subtraction. Sometimes factoring was helpful there.

For matrix multiplication, you multiply the row elements by the column elements and add them up to get the products elements. Sometimes if the dimensions of the matrices do not match up correctly, you cannot multiply them.

Compound interest and half lives – tutoring precalculus

We looked at problems with continuous interest in terms of banking and similar problems for elements with radioactive decay. The same equation can basically be used for both situations, you can change letters if it seems to make more sense that way.

The lnx and e^x functions can counteract each other, much like arcsinx and sinx.

There is another equation for half life specifically, but it’s easy enough to get to an equation with that idea using the original equation and therefore not memorizing more equations than necessary.