Tutoring Precalculus, Log function tricks

We started with some problems involving logarithms and natural logarithms.

By acting the log function on something with an exponent, the exponent can be brought in front.

Continuous compound interest uses a function with the natural number e to the power of the rate multiplied by time.

Also went over test corrections.

One trick was divide both sides of an equation by two to simplify everything.

Vertex form of quadratic equation, Tutoring Precalculus

We started by looking at some problems with division of polynomials. Chose to use synthetic division, long division would be an option also.

To fully factor, for example, a fourth degree polynomial, if it comes out without a remainder then the combination of all factors is the fully factored form. Like you could factor 12 as 2*3*2.

Another problem seemed to call for the vertex form of a quadratic equation y = a(x-h)^2 + k where (h, k) is the vertex. Then you could convert that to standard form.

Talked a bit more about imaginary numbers.

“exact value of cos 255 degrees without a calculator” Tutoring Precalculus

We started with some problems involving bearings. The direction can get mixed up pretty easily, but the problems were fairly straightforward.

Another problem asked for the exact value of cos 255 degrees without a calculator. It seemed like a half-angle formula might be a good approach.

One problem involved area which meant needing to know the area formula for a triangle 1/2 bh.

And using inverse trig functions was useful. Other problems in the review did require the law of cosines and the law of sines.

Tutoring Precalculus: Bearing Measurement, Unit Vectors, Using the Law of Cosines

We first looked at getting a bearing measurement between three cities.

The law of cosine was used. As more information is found, you can use the Law of Sine and properties of triangles.

Another problem had a unit vector in which the magnitude was 1.

And other problems involved adding vectors which meant sometimes multiplying them by coefficients and then lining them up head to tail.

Why to use the Law of Sines and the Law of Cosines

We looked at problems involving the Law of Sines and the Law of Cosines. You can use these equations with any triangle, they do not have to be right triangles. Depending on the information you have, you use one or the other.

You’ll have two sides and an angle generally or two angles and a side. And as you fill in more information you may want to switch to the Law of Sines since it tends to be less work.

And if you have two angles, you can just subtract them from 180 degrees to get the third angle. The law of cosines is really like a more general form of the Pythagorean Theorem, it has a similar function but does not require the triangle to be a right triangle.

Using memory in your calculator can be helpful for these problems as well since you often need to combine several numbers that are not whole numbers.

sin^2x = (sinx)^2 Squared Trig Function Notation

We went into verifying some equations with trigonometric. They involved angles in both degrees and radians.

Sometimes with equations that had notation of something like sin^2x, which means (sinx)^2 but is the more common notation.

No calculators used in the problems, so the angles all were variations of 45-45-90 or 30-60-90.

Solving Inequalities and finding intervals, Tutoring Precalculus

We started by looking at the Friday Sheet.

With inequalities, the steps are basically

1. Use factoring/etc to find boundary points
2. Plot the boundary points with open or closed circles on a number line
3. Test a point in each of the three regions

Spent a bit of time plotting graphs. Even if you don’t really know the shape, you can plot points. It helps quite a bit to know the shape in advance though.

Reviewed the equation for the vertex of a parabola. x = -b/2a. And also reviewed where you see all of those letters in the general quadratic equation.

Getting Arc Length, Tutoring Precalculus

We started with a word problem trying to find arc length based on latitude and longitude.

Reviewed the equations for the circumference of a circle and the area of a circle.

You can get arc length by multiplying the angle (in radians) by the radians. Thinking of the equation for circumference can help for that since it’s the angle of an entire circle in radians multiplied by the radius. Or you can multiply the fraction of the entire circle by the circumference.

Looked at a few more complicated graphing problems of trig functions.

Variations in plotting trig functions, tutoring Precalculus

We looked at graphing different trigonometric functions with variations.

Started with sin x and cos x. Looked at expansions/contractions, shifts vertically and horizontally.

Used the chocolate trick to remember the less common trig functions

Vertical asymptotes are also useful to plot when they appear.

How Chocolate Can Help You Learn Trigonometric Functions cosecant, secant, cotangent


Why would chocolate help you with the trig functions?

If you know SOHCAHTOA, which you should learn if you are taking math, you know the ratios for three trig functions.

There are three other functions that have the reciprocals of the ratios of those functions.

I remember mixing up secant and cosecant sometimes, which one was paired with sine and which with cosine.

If you know SOHCAHTOA, you could learn CHOSHACAO. But more simply than that, if you know SOHCAHTOA, you can just remember Chocolate.

CHO is the first part and if you line them up like this:



The first of the lesser known trig functions lines up with Sine. So the ratio for cosecant is the reciprocal of the ratio for sine.

It’s fairly easy to remember that cotangent and tangent line up. Therefore you now know two of three so secant has to line up with cosine.