Monday 24 April class

Posted on April 24, 2017 by Sybil Shaver

Topics:

• Please make sure that you review the material on Trigonometry from the last two classes. You must become very familiar with the two special right triangles, and with the standard position for a triangle in the coordinate plane.

• Reflecting a right triangle in standard position: (this was done last time, but I’m mentioning it here because we make use of it today)

We can take a right triangle in standard position and reflect it over the x-axis or over the y-axis or both, to get three other triangles in the other three quadrants. These triangles are not in standard position, but they do have one vertex at the origin, and their right angles are on the x-axis (although not necessarily on the positive x-axis). We are going to make use of these reflected triangles in studying trig on the coordinate plane: they will be called reference triangles.

You can see some reflected right triangles in this Math is Fun post on the coordinate plane.

We reflected the 5-12-13 right triangle into the four quadrants and found the coordinates of the vertex which is not on an axis in each case. The coordinates of all four points are the same except for their signs, which depend on the quadrant.

• Angles in standard position:

An angle in the coordinate plane is said to be in standard position if its vertex is at the origin, and its initial side is on the positive x-axis. (Remember that we think of angles as rotations!) The angle is positive if the rotation is counter-clockwise, and the angle is negative if the rotation is clockwise.

• Coterminal angles:

Angles in standard position are called coterminal if they have the same terminal side. Coterminal angles differ by adding or subtracting a full rotation (360 degrees, or 2\pi) one or more times. From the definitions of the trig functions in the coordinate plane, it is clear that coterminal angles will have the same values for the trig functions. This shows that the trig functions are periodic.

• Radian measure of angles:

See the Math is Fun webpage on this topic and more discussion of radians, and also their exercises which are at the bottom of that second link! Very useful!

• The coordinate plane definitions of the trig functions: 

These are given in the outline notes that I handed out in class. This handout is also available here:

MAT1275TrigFns3definitions

Also see the Math is Fun post on the coordinate plane.

• The signs of the trig functions in the four quadrants: (the Math is Fun post also discusses these)

These are easy to see from the coordinate plane definitions, since the sign of the sine will be the same as the sign of the second coordinate b, and the sign of the cosine will be the same as the sign of the first coordinate a. (Notice that I am calling the two coordinates a and b, because we will be using x to stand for the angle later on!) We can then get the sign of tangent by using the identity \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}.

I do not advocate using mnemonic devices to remember the signs, but rather, think about the signs of the coordinates in the four quadrants and how the coordinates are connected to the trig functions. This is a much stronger way to get all of those concepts connected in your memory!

The importance of knowing the signs of the trig functions in the four quadrants is that the trig functions of any angle are the same as the trig functions of its reference angle (which we will discuss next time), except for the sign.

Also, if we know one trig function of an angle and we know which quadrant the angle is in, we can compute the values of all the trig functions for that angle. We did two examples of this in class: here is one.

Example: If \tan(\theta) = -\frac{3}{4} and $\cos(\theta) < 0$, what are the values of the other five trig functions?

 

• The unit circle definitions of the trig functions:

This is just a version of the coordinate plane definitions, where we take the radius  r to be 1. These definitions are also given in the handout.

• Very important identities which are easy to prove using the unit circle definitions:

\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}

\sin^{2}(\theta) + \cos^{2}(\theta) = 1 (The Pythagorean Identity)

There are two more versions of the Pythagorean Identity that are easy to derive from this one: we will discuss them later.

Finding values of the trig functions for angles related to the special right triangles, using the unit circle definitions and reflecting the special right triangles.

You should practice until you are able to come up with the values of the trig function for these angles very quickly.

 

Homework:

• Review the material, definitions and vocabulary, and the examples discussed in class. Make sure that you know all the vocabulary, and the three definitions of the trig functions. Follow the links in the notes above to help you review!

• Do the WeBWorK: not due until Sunday evening because of Test 3, but don’t delay! There are 3 parts (so far).

• Also do the following problems:

  1. Draw a right triangle whose legs are 3 and 4 in standard position in the coordinate plane, so the side of length 3 is along the x-axis. Then reflect that triangle over the x-axis and the y-axis and both axes to get three other triangles in the three other quadrants. Find the coordinates of the vertices which are not on the axes for those four triangles.
  2. Find the values of the six trig functions if \tan(\theta) = \frac{4}{3} and \sin(\theta) < 0. Simplify them completely.
This entry was posted in Uncategorized. Bookmark the permalink.