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.


What is the sin of 90 degrees, Tutoring Precalculus

We looked at coordinates and their rays from the origin and the six trigonometric functions. To do that, we used SOHCAHTOA and to remember the other three functions, you can use the word ‘chocolate’ since CHO, cscx=H/O is the reciprocal of sinx.
To visualize the ‘triangles’ for the two directions on the x-ax and the two directions on the y-axis, I’ll draw something like 85 degrees instead of 90 since it’s fairly close and then label one side of the triangle as zero and the other two as equal values (1 usually works well).


Tutoring Precalculus, Focusing on the Unit Circle and 30-60-90 and 45-45-90 Triangles

We started by focusing on the unit circle. There are two main triangles that form the basis for a lot of what they are doing in the class. 30-60-90 and 45-45-90. Knowing the ratios of the sides for those triangles allows you to more easily do much of the work. So we looked at those triangles in various positions.


Knowing what sinx and cosx look like graphed can be useful as well.

We also looked at how triangles with 85 degrees and 5 degrees are similar to 90 and 0 degrees and how that can be useful.

Tutoring Precalculus, Trigonometric Functions and Properties

We started by looking at a problem with triangles. Two smaller triangles made up a larger triangle. The larger triangle was a right triangle. Basically, we started finding different angles, and use the law of sines at one point. Then we could use a trigonometric identity.

For another problem, it helped to create a new variable.

We spent a little time on graphs of trigonometric identities and also understanding the domain and range. That included looking at the amplitude and expansion and compression of functions.