×

### Let's log you in.

or

Don't have a StudySoup account? Create one here!

×

### Create a StudySoup account

#### Be part of our community, it's free to join!

or

##### By creating an account you agree to StudySoup's terms and conditions and privacy policy

Already have a StudySoup account? Login here

by: devin mills

18

2

9

# math 111 trig function MATH 111A - 01

devin mills
UCSC

### Preview These Notes for FREE

Get a free preview of these Notes, just enter your email below.

×
Unlock Preview

### Preview these materials now for free

Why put in your email? Get access to more of this material and other relevant free materials for your school

trig function
COURSE
Algebra
PROF.
Mason,G.
TYPE
Class Notes
PAGES
9
WORDS
KARMA
25 ?

## Popular in Department

This 9 page Class Notes was uploaded by devin mills on Saturday June 4, 2016. The Class Notes belongs to MATH 111A - 01 at University of California - Santa Cruz taught by Mason,G. in Spring 2016. Since its upload, it has received 18 views.

×

## Reviews for math 111 trig function

×

×

### What is Karma?

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 06/04/16
Review : Trig Functions The intent of this section is to remind you of some of the more important (from a Calculus  standpoint…) topics from a trig class.  One of the most important (but not the first) of these  topics will be how to use the unit circle.  We will actually leave the most important topic to the  next section.   First let’s start with the six trig functions and how they relate to each other.       Recall as well that all the trig functions can be defined in terms of a right triangle.   From this right triangle we get the following definitions of the six trig functions.     Remembering both the relationship between all six of the trig functions and their right triangle  definitions will be useful in this course on occasion.   Next, we need to touch on radians.  In most trig classes instructors tend to concentrate on doing  everything in terms of degrees (probably because it’s easier to visualize degrees).  The same is  true in many science classes.  However, in a calculus course almost everything is done in  radians.  The following table gives some of the basic angles in both degrees and radians.   Degree 0 30 45 60 90 180 270 360 Radians 0   Know this table!  We may not see these specific angles all that much when we get into the  Calculus portion of these notes, but knowing these can help us to visualize each angle.  Now, one more time just make sure this is clear.   Be forewarned, everything in most calculus classes will be done in radians!   Let’s next take a look at one of the most overlooked ideas from a trig class.  The unit circle is  one of the more useful tools to come out of a trig class.  Unfortunately, most people don’t learn it as well as they should in their trig class.    Below is the unit circle with just the first quadrant filled in.  The way the unit circle works is to  draw a line from the center of the circle outwards corresponding to a given angle.  Then look at  the coordinates of the point where the line and the circle intersect.  The first coordinate is the  cosine of that angle and the second coordinate is the sine of that angle.  We’ve put some of  the basic angles along with the coordinates of their intersections on the unit circle.  So, from the  unit circle below we can see that    and  .    Remember how the signs of angles work.  If you rotate in a counter clockwise direction the angle is positive and if you rotate in a clockwise direction the angle is negative.   Recall as well that one complete revolution is  , so the positive x­axis can correspond to  either an angle of 0 or   (or  , or  , or  , or  , etc. depending on the direction of rotation).  Likewise, the angle   (to pick an angle  completely at random) can also be any of the following angles:    (start at   then rotate once around  counter clockwise)    (start at   then rotate around  twice counter clockwise)    (start at   then rotate once  around clockwise)    (start at   then rotate around  twice clockwise)   etc.   In fact   can be any of the following angles   In this case n is the number of  complete revolutions you make around the unit circle starting at  .  Positive values  of n correspond to counter clockwise rotations and negative values of n correspond to clockwise  rotations.   So, why did I only put in the first quadrant?  The answer is simple.  If you know the first  quadrant then you can get all the other quadrants from the first with a small application of  geometry.  You’ll see how this is done in the following set of examples.   Example 1  Evaluate each of the following. (a)   and     [Solution ] (b)   and    [Solutio n] (c)   and    [Solution] (d)    [Solution] Solution (a) The first evaluation in this part uses the angle  .  That’s not on our unit circle  above, however notice that  .  So   is found  by rotating up   from the negative x­axis.  This means that the line for   will be a mirror image of the line for   only in the second quadrant.  The coordinates for   will be the coordinates for   except the x coordinate will be negative.   Likewise for   we can notice that  , so this angle can be found by rotating down   from the negative x­axis.   This means that the line for   will be a mirror image of the line for   only in the third quadrant and the coordinates will be the same as the coordinates for   except  both will be negative.   Both of these angles along with their coordinates are shown on the following unit circle. From this unit circle we can see that    and  .   This leads to a nice fact about the sine function.  The sine function is called an odd function  and so for ANY angle we have [Return to Problems]   (b) For this example notice that   so this means we  would rotate down   from the negative x­axis to get to this angle.  Also   so this means we would rotate up   from the negative x­axis to get to this angle.  So, as with the last part, both of these angles  will be mirror images of   in the third and second quadrants respectively and we can use  this to determine the coordinates for both of these new angles.   Both of these angles are shown on the following unit circle along with appropriate coordinates  for the intersection points.   From this unit circle we can see that    and  .  In this case the cosine  function is called an even function and so for ANY angle we have . [Return to Problems]   (c) Here we should note that   so    and   are in fact the same angle!  Also note that this angle will be the mirror image  of   in the fourth quadrant.  The unit circle for this angle is   Now, if we remember that   we can  use the unit circle to find the values of the tangent function.  So,   .   On a side note, notice that   and we can see that the  tangent function is also called an odd function and so for ANY angle we will have . [Return to Problems]   (d) Here we need to notice that  .  In other  words, we’ve started at   and rotated around twice to end back up at the same point on  the unit circle.  This means that                                             Now, let’s also not get excited about the secant here.  Just recall that                                                             and so all we need to do here is evaluate a cosine!  Therefore,

×

×

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

×

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

Steve Martinelli UC Los Angeles

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

Kyle Maynard Purdue

#### "When you're taking detailed notes and trying to help everyone else out in the class, it really helps you learn and understand the material...plus I made \$280 on my first study guide!"

Bentley McCaw University of Florida

Forbes

#### "Their 'Elite Notetakers' are making over \$1,200/month in sales by creating high quality content that helps their classmates in a time of need."

Become an Elite Notetaker and start selling your notes online!
×

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.