Link to the take-home section of Midterm 2, due by the end of class on Wed., May 6

Here is the link to the take-home section.

Homework 10 is due by midnight tonight.

In class, we will discuss changing the formulas for circles in Cartesian coordinates into  polar coordinates.

The in-class section will be given out at 11:00 am on Wednesday, due by 12:15 pm, which is also the deadline for the take-home section.

Here is a link to today's notes.

Notes for Homework #12, due Tues. Apr. 30 NOT ACCEPTED LATE

Notes for converting polar to Cartesian and vice versa

Note for plotting polar coordinate functions

The website Wolfram Alpha is a good resource for plotting. For example, use the instruction

plot r = cos(theta)

To see an accurate picture of a circle with its center at (= ½, 0) and radius = ½, passing through both (0, 0) and (1, 0)

Note: The x-axis is all points of the form (x, 0) in Cartesian coordinates. The form looks exactly the same in Polar coordinates, since the angle 0 is the x-axis.

Notes for Homework #11, due Tues., April 23

Notes for conversion from polar coordinates to Cartesian coordinates and vice versa.


 

Polar coordinates.

In class Monday, we introduced the idea of polar coordinates.  In rectilinear coordinates (x, y), the first number tells us how far right (positive) or left (negative) we move on the horizontal axis and the second number tells us how far up (positive) or down (negative) we move on the vertical axis.

In polar coordinates we have (r, theta).  There is a central point, not unlike (0, 0) in the xy-axis system, and we consider that we are pointing in a direction we call angle 0 in radians.  The standard is to make that direction to the right, the same as the positive x-axis in rectilinear.  r is the distance from the origin and theta is the angle given in radians.

It is acceptable to have negative values for distance and for angles.

Unlike rectilinear coordinates, polar coordinates are not unique.  The easiest example of this are the coordinate pairs (0, 0) and (0, 1).  The first zero means we are a distance of 0 from the origin.  That means we are on the origin.  The angle we turn doesn't change where we are.

Even when points aren't the origin, the polar coordinates are not unique. The simplest example is

(1, 0) = (1, 2pi)

What this says is if we are 1 away from the origin and pointing to the right, this is the same as being 1 away from the origin and turning a full circle. In fact, adding or subtracting any multiple of 2pi to an angle brings us back to pointing in the exact same direction.

We will be working more in polar coordinates for the rest of the week, possibly longer.

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The trig functions all around the unit circle

In this picture, the angle in the first quadrant closes to the x-axis is labeled a°. It is the upper right hand point of the red rectangle.  (90 - a)° is also in the first quadrant and is the upper right hand corner of the blue rectangle. These two angles are complementary.  The other six angles are some "nice" angle away from a° or (90 - a)°, adding either 90°, 180° or 270°.  All the trig values of sine, cosine and tangent for the eight angles labeled here can be derived from the three values sina, cosa and tana.  We already seen the relationship between the trig values of complementary angles.

sin(90 - a)° = cosa
cos(90 - a)° = sina
tan(90 - a)° = 1/tana

When 90° is added, sine and cosine switch values and the cosine of the new angle is negated as follows.

sin(90 + a)° = cosa
cos(90 + a)° = -sina
tan(90 + a)° = -1/tana

90° more than the complementary angle is the same story with those values.


sin(180 - a)° = sina
cos(180 - a)° = -cosa
tan(180 - a)° = -tana

Adding 180° puts us at the antipode, the polar opposite of where we were.  Sine and cosine will just be negated, while tangent will remain exactly the same.


sin(180 + a)° = -sina
cos(180 + a)° = -cosa
tan(180 + a)° = tana

180° beyond the complement is a similar story.


sin(270 - a)° = -cosa
cos(270 - a)° = -sina
tan(270 - a)° = 1/tana

And then we have adding 270°. This is like adding 180° to the values we got when adding 90°.


sin(270 + a)° = -cosa
cos(270 + a)° = sina
tan(270 + a)° = -1/tana

270° more than the complementary angle is the same story with those values.


sin(360 - a)° = -sina
cos(360 - a)° = cosa
tan(360 - a)° = -tana

Example: 60° is one of our "famous" angles.

sin60° = sqrt(3)/2
cos60° = 1/2
tan60° = sqrt(3)

Find the following values using these three pieces of information.

a) sin150°
b) cos240°
c) tan330°
d) tan 120°
e) sin 210°
f) cos 300°

Answers in the comments.