PRE-CALCULUS: Quarter 2 - Module 1

ANGLES IN A UNIT CIRCLE

Pre-Calculus: Quarter 2 - Module 1

Objectives

• illustrate the unit circle and its relationship between linear and angular measures of arcs in a unit circle; and 

• convert degree measure to radian measure and vice versa

PART 1

Angles in a Unit Circle

Open Ended

How angles are formed? What do you understand about angles?

Euclidean Geometry vs. Trigonometry

Angles in trigonometry differ from angles in Euclidean geometry in the sense of motion. An Angle in geometry is defined as a union of rays (that is, static) and has a measure between 0° and 180°. An angle in trigonometry is a rotation of a ray, and, therefore, has no limit. It has positive and negative directions and measures (Garces, I. J.,2016). 

Euclidean Geometry - "Junior High Geom."

Angles in "degree" measure

Trigonometry - Angles plotted in a plane can be negative

Angles can be in "degree" and "radian" measure

Angles can be positive or negative

ANGULAR MEASURE: Degree and Radian

How radians related to degrees?

[Please watch my video on "Intro to Radian Measure" posted on our Facebook Page before proceeding to the next slide]

PART 2

Converting Radian Measure to Degree Measure

and Vice Versa

Please complete the table (next slide) first using your calculator.

Then, answer the questions to follow.

(Energizer)

Multiple Choice

If 180° is half of a circle, how many radians are there?

• $2\pi$ rad

• $\pi$ rad

• $\frac{1}{4}\pi$ rad

Multiple Choice

If 90° is a quarter of a circle, how many radians are there?

•  $\frac{1}{2}\pi\$  rad

•  $\frac{3}{4}\pi$ rad

•  $\frac{1}{7}\pi$  rad

Multiple Choice

If 270° is three-quarters of a circle, how many radians are there?

•  $\frac{1}{2}\pi\$  rad

•  $\frac{3}{4}\pi$ rad

•  $\frac{3}{2}\pi$  rad

Multiple Choice

If 45° is __________________of a circle, so there are _______ radians. 

• one-eight ; $\frac{1}{8}\pi\$ rad 

• one-eight ; $\frac{1}{4}\pi$ rad

• one-fourth ; $\frac{1}{4}\pi$  rad

To convert degree measure to radian measure: multiply the given measurement to to get the desired radian measure

 $\left(\frac{\pi\ rad}{180\degree}\right)$  

see example next slide...

Write  $45\degree$  in radians

In the example, we multiply the given  $45\degree$  to  $\left(\frac{\pi\ rad}{180\degree}\right)$  then cancel out the degree symbol. Next is  $\frac{45\ \cdot\pi\ rad}{180}=\frac{1}{4}rad$  

Express 75° in radians

Multiply the given 75° to $\left(\frac{\pi\ rad}{180\degree}\right)$ . Then divide both 75 and 180 with their greatest common factor (or just simple 75 divided 180). So,   $\frac{75\degree\cdot\pi\ rad}{180\degree}=\frac{5\pi}{12}rad$  

Convert

 $240\degree$  to radian measure

More Examples

Your turn!

Multiple Choice

Convert  $120\degree\$  to radian measure.

•  $\frac{1}{3}\pi\ radians$  

•  $\frac{2}{3}\pi\ radians$  

•  $\frac{1}{6}\pi\ radians$  

Multiple Choice

Convert  $-30\degree\$  to radian measure.

•  $\frac{1}{3}\pi\ radians$  

•  $\frac{2}{3}\pi\ radians$  

•  $-\frac{1}{6}\pi\ radians$  

Multiple Choice

Convert  $225\degree\$  to radian measure.

•  $\frac{2\pi}{3}\ radians$  

•  $\frac{4\pi}{3}\ radians$  

•  $\frac{5\pi}{4}\ radians$  

Multiple Choice

Convert 110° 50´ 30" to radians

(tip: simplify the given to degree measure first)

•  $\frac{31}{50}\pi\ radians$  

•  $\frac{40}{51}\pi\ radians$  

•  $\frac{53}{49}\pi\ radians$  

To convert radian measure to degree measure: multiply the given measurement to

 $\left(\frac{180\degree}{\pi\ rad}\right)$  

see examples next slides...

Convert $\frac{5}{9}\pi\$ radians to degree measure?

Multiply the given  $\frac{5}{9}\pi\ rad\ to\ \frac{180\degree}{\pi\ rad}.\ So,\ \left(\frac{5}{9}\pi\ rad\right)\left(\frac{180\degree}{\pi\ rad}\right)\ =\left(\frac{5\cdot180\degree}{9}\right)=100\degree$  Note that we cancel out  $\pi\ rad$  .

Multiple Choice

Convert $\frac{4}{3}\pi\ radians\$ 

to degrees.

•  $240\degree$  

•  $360\degree$  

•  $657\degree$  

Multiple Choice

Convert $\frac{7}{4}\pi\ rad$ 

to degrees.

•  $365\degree$  

•  $450\degree$  

•  $315\degree$  

Multiple Choice

Convert $-\frac{7}{6}\pi\ rad$ 

to degrees.

•  $204\degree$  

•  $210\degree$  

•  $222\degree$  

YOURT TURN!

So, in degree-radian conversion, we must note ...

->  to multiply the given radian measure with $\left(\frac{180\degree}{\pi\ rad}\ \right)$ to get the degree measure

-> to multiply the given degree measure with  $\left(\frac{\pi\ rad}{180\degree}\right)$  to get the radian measure

Try this little Extra!

Multiple Choice

Which of the following arc length is the shortest?

• 
  

  $\frac{5}{6}\pi\$
  

• $\frac{5}{4}\pi\$

• $\frac{5}{3}\pi\$

• $\frac{5}{2}\pi\$

Multiple Choice

Which of the following radian measures is greater than 180°?

• 
  

   $\frac{\pi}{6}\$ 
  

•  $\frac{\pi}{2}\$ 

•  $\frac{5\pi}{3}$ 

•  $\frac{7\pi}{8}\$ 

Multiple Choice

Which of the following points is on the unit circle? 

• 
  

   $\left(-1,0\right)$  
  

•  $\left(1,1\ \right)$ 

•  $\left(2,0\right)$ 

•  $\left(0,0\right)$ 

Multiple Choice

How many equal arcs is the circle to be divided if each arc measures  $\frac{\pi}{40}$   

• 
  

  50

• 60

• 70

• 80

Multiple Choice

What is the value of  $\theta$  in radian measure?

•  $\frac{2\pi}{6}\ rad$  
  

  
  

•  $\frac{\pi}{5}\ rad$  
  

•  $\frac{\pi}{6}\ rad$  
  

•  $\pi\ rad$  
  

Pick one item from the left and one from right. Convert them to degrees. Answer is on the next slide.

answers:

 $\frac{\pi}{4}\ rad=45\degree$           $-\frac{\pi}{4}\ rad=-45\degree$   

 $\frac{\pi}{6}\ rad\ =30\degree$          $-\frac{\pi}{6}\ rad\ =-30\degree$  

 $\frac{\pi}{2}rad\ =90\degree$            $-\frac{\pi}{2}rad\ =-90\degree$  

I hope you learn something. See you!