
\documentclass[12pt]{article}
\begin{document}
\def\vskp{\vskip .125truein}

\def\Lvskp{ \vskip .125 truein }
\def\Bvskp{ \vskip 1.05 truein }
\def\LINE {\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}

\noindent
Math 12 (Spring 2001)

\noindent
Kouba

\noindent
Exam 3
\Bvskp

\noindent
Please PRINT your name here : \LINE
\Lvskp

\noindent      
Your HW/Exam ID Number \_\_\_\_\_\_\_\_\_\_\_\_
\Lvskp
\Lvskp
\Lvskp

1.   PLEASE DO NOT TURN THIS PAGE UNTIL TOLD TO DO SO.
\Lvskp

2.   IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO, IN ANY WAY, ASSIST ANOTHER PERSON IN THE COMPLETION OF THIS EXAM.  PLEASE KEEP YOUR OWN WORK COVERED UP AS MUCH AS POSSIBLE DURING THE EXAM SO THAT OTHERS WILL NOT BE TEMPTED OR DISTRACTED.  THANK YOU FOR YOUR COOPERATION.
\Lvskp

3.  YOU WILL NEED A CALCULATOR ON THIS EXAM.
\Lvskp

4.  No notes, books, or classmates may be used as resources for this exam.  
\Lvskp

5.  Read directions to each problem carefully.  Show all work for full credit.  In most cases, a correct answer with no supporting work will NOT receive full credit.  What you write down and how you write it are the most important means of your getting a good score on this exam.  Neatness and organization are also important.
\Lvskp

6.  Put units on answers where units are appropriate.
\Lvskp

7.  You have until 9:50 a.m. sharp to finish the exam. 
\Lvskp
8.  Make sure that you have 7 pages including the cover page.
\Lvskp \Lvskp \Lvskp
TRIGONOMETRY IDENTITIES ARE ON THE NEXT PAGE.

\vfill\eject

TRIGONOMETRY IDENTITIES

\Lvskp \Lvskp

1. $ \sin( \alpha+\beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta $
\Lvskp
2. $ \sin( \alpha-\beta) = \sin\alpha \cos\beta - \cos\alpha \sin\beta $
\Lvskp
3. $ \cos( \alpha+\beta) = \cos\alpha \cos\beta - \sin\alpha \sin\beta $
\Lvskp
4. $ \cos( \alpha-\beta) = \cos\alpha \cos\beta + \sin\alpha \sin\beta $
\Lvskp
5. $ \tan( \alpha+\beta) = \displaystyle{ \tan\alpha + \tan\beta \over 1 - \tan\alpha \tan\beta } $
\Lvskp
6. $ \tan( \alpha-\beta) =  \displaystyle{ \tan\alpha - \tan\beta \over 1 + \tan\alpha \tan\beta }  $
\Lvskp
7. $ \sin( 2\alpha) = 2\sin\alpha \cos\alpha  $
\Lvskp
8. $ \cos( 2\alpha) = \cos^2\alpha - \sin^2\alpha = 2\cos^2\alpha - 1 
= 1 - 2\sin^2\alpha $
\Lvskp
9. $ \tan( 2\alpha) =  \displaystyle{ 2\tan\alpha  \over 1 - \tan^2\alpha }  $
\Lvskp
10. $ \sin( \alpha/2) = \pm \sqrt{ 1- \cos\alpha \over 2 }$
\Lvskp
11. $ \cos( \alpha/2) = \pm \sqrt{ 1+ \cos\alpha \over 2 }$
\Lvskp
12. $ \tan( \alpha/2) =  \displaystyle{ \sin\alpha \over 1 + \cos\alpha  }  $









\vfill\eject


\noindent
1.)  (4 pts. each)  Assume that $ \pi/2 < \alpha < \pi $ and $ \tan \alpha = -3/4 $.  Find exact values for each by drawing an appropriate circle and triangle.  Use the identities on page 2.
\Lvskp
a.) $ \sin \alpha $
\Lvskp \Lvskp \Lvskp
b.) $ \cos \alpha $
\Lvskp \Lvskp
c.) $ \sin(2 \alpha) $
\Lvskp \Lvskp \Lvskp
d.) $ \cos(2 \alpha) $
\Lvskp \Lvskp \Lvskp
e.) $ \tan(2 \alpha) $
\Lvskp \Lvskp \Lvskp
f.) $ \sin( \alpha/2) $
\Lvskp \Lvskp \Lvskp
g.) $ \cos( \alpha/2) $
\Lvskp \Lvskp \Lvskp
h.) $ \tan( \alpha/2) $

\Lvskp \Lvskp \Lvskp \Lvskp \Lvskp
 

\noindent
2.)  (4 pts. each)  Draw an appropriate circle and triangle, then use a reference angle to get the exact value of each
\Lvskp
a.) $ \sin 135^o $
 
\Bvskp
b.) $ \cos 210^o $

\vfill\eject



\noindent
3.)  (6 pts.) Use the fact that $ 15^o = 45^o - 30^o $ to find the exact value of 
$ \tan 15^o $.

\Bvskp \Bvskp 

\noindent
4.) (5 pts. each)  Convert the following. Give exact answers.  No calculators.   
 \Lvskp
a.) $ 45^o $ to radians
 
\Bvskp  \Lvskp  \Lvskp

b.) $ 5\pi/6 $ radians to degrees

\Bvskp  \Lvskp  \Lvskp


\noindent
5.) (6 pts.)  Find the exact area of the given triangle.
\Lvskp


\vfill\eject



\noindent
6.) (6 pts.)  When a clock reads 10:00, what is the exact radian measure of the (smaller) angle between the hour hand and the minute hand.
\Lvskp \Lvskp \Lvskp
\Bvskp

\noindent
7.) (6 pts. each)  Consider the given circular sector.
\Lvskp
  a.)  Find the area of the sector rounded 

\noindent  
to two decimal places.
\Bvskp \Lvskp \Lvskp  \Lvskp \Lvskp

  b.)  Find the perimeter of the sector rounded to two decimal places.  
\Bvskp \Lvskp \Lvskp  \Lvskp \Lvskp


\noindent
8.) (6 pts.) Use a calculator to solve $ 4 \sin \theta = 3 $ for $ 0 < \theta < 2 \pi $.  Round answers to two decimal places.





\vfill\eject

\noindent
9.)  (7 pts.)  Prove that the following equation is an identity.
\Lvskp
   $ \displaystyle{ { \sin ( 2 \theta ) \over \sin \theta } -
 { \cos ( 2 \theta ) \over \cos \theta }= \sec \theta } $ 

\Bvskp
\Bvskp
\Bvskp \Lvskp \Lvskp \Lvskp \Lvskp

\noindent
10.)  A wheel of radius 5 feet is spinning at the rate of 10 revolutions per second.

\Lvskp
  a.) (3 pts.)  Find the angular velocity of the wheel in degrees per second.  


\Bvskp \Lvskp \Lvskp  \Lvskp \Lvskp

  b.) (4 pts.)  Find the linear velocity of a point on the circumference of the wheel in feet per second.   Round your answer to two decimal places.  

\vfill\eject 



\noindent
The following EXTRA CREDIT PROBLEM is worth 10 points.  This problem is OPTIONAL.
\Lvskp
\noindent
1.) The figure shows a circular sector with radius $ r $ cm. and central angle $ \theta $ radians.  The perimeter of the sector is 12 cm.  Express $ r $ as a function of $ \theta $.





\end{document}
