Trigonometry

Trigonometry

Radian Conversion

Radian - another unit used to measure angles.

pi is an example of an angle in radians.

 $\pi^c=180\degree\ \ \ 2\pi^c=360\degree$  

Converting Degrees to Radians

 $180\degree=\pi^c$  
 $1\degree=\frac{\pi}{180}rad$  

(a)  $120\degree$                               (b)  $45\degree$                            (c)  $90\degree$  
 $1\degree=\frac{\pi}{180}rad$                       $1\degree=\frac{\pi}{180}rad$                  $1\degree=\frac{\pi}{180}rad$  
 $\therefore120\degree=\frac{\pi}{180}\times120$     $\therefore45\degree=\frac{\pi}{180}\times45$      $\therefore90\degree=\frac{\pi}{180}\times90$  
 $=\frac{2\pi}{3}or\frac{2}{3}\pi^c$                       $=\frac{\pi}{4}or\frac{1}{4}\pi^c$                      $=\frac{\pi}{2}or\ \frac{1}{2}\pi^c$  

Converting Radians to Degrees

 $\pi=180\degree$  
 $1^c=\frac{180}{\pi}$  

(a)  $\frac{3\pi}{2}$                               (b)  $\frac{\pi}{18}$                                (c)  $\frac{2\pi}{15}$  
 $1^c=\frac{180}{\pi}$                                  $1^c=\frac{180}{\pi}$                        $1^c=\frac{180}{\pi}$  
 $\therefore\frac{3\pi}{2}=\frac{180}{\pi}\times\frac{3\pi}{2}$         $\therefore\frac{\pi}{18}=\frac{180}{\pi}\times\frac{\pi}{18}$              $\therefore\frac{2\pi}{15}\times\frac{180}{\pi}$  
 $=270\degree$                                   $=10\degree$                                $=24\degree$  

Length of Arc

Formula :  $s=r\theta$  

where:
*theta is in radians 
*s is the length of the arc
*r is the radius 

Example 1

 $r=12cm$  
 $\theta=\frac{\pi^c}{4}$  

 $s=r\theta$  
 $\therefore s=12\left(\frac{\pi^c}{4}\right)$  
 $=9.4cm$  

Example 2

 $r=9mm$  
 $\theta=\frac{5\pi^c}{6}$  

 $s=r\theta$  
 $\therefore s=9\left(\frac{5\pi}{6}\right)$  
 $=23.6mm$  

Example 3

 $180\degree=\pi$  
 $1\degree=\frac{\pi}{180}$  
 $\therefore35\degree=\frac{\pi}{180}\times35$  
 $=\frac{7\pi^c}{36}$  

 $r=10\ \ \ \theta=\frac{7\pi^c}{36}$  
 $s=r\theta$  
 $s=10\times\frac{7\pi}{36}$  
 $=6.1cm\ \left(1d.p\right)$  

Example 4

 $360-97=263\degree$  
 $1\degree=\frac{\pi}{180}$  
 $\therefore263=\frac{\pi}{180}\times263$  
 $=\frac{263}{180}\pi$  

 $s=r\theta$  
 $=8.5\times\frac{263}{180}\pi$  
 $=39cm$  

Example 5

A diagram shows a sector of a circle center O. The radius of the circle is 6cm. Angle AOB.
(a) Draw a diagram to represent this.
(b) Determine the perimeter of the sector leaving he answer in terms of  $\pi$  . 

(b)  $1\degree=\frac{\pi}{180}$  
 $\therefore120=\frac{\pi}{180}\times120$  
 $=\frac{2}{3}\pi$  

 $s=r\theta$  
 $=6\left(\frac{2\pi}{3}\right)=4\pi$  

 $\therefore Perimeter=4\pi+6+6=\ 4\pi\ +12$