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TRIGONOMETRY : LESSONS 2, 3, 4, 5 and 6

LESSON 2

TRIGONOMETRIC EQUATIONS

Example. 1. Find the solution of the equation  2 sin x – 1 = 0.

                          2 sin x – 1 = 0
 
                          2 sin x = 1

                             sin x =  ½

                                x =  30°  or  150°

                 2.  2 sin x + 1 = 0

                3.  tan x  + 1  = 0

                4.  tan x  =  1

 When  equations are factorable, factor the equation, equate factors to zero and solve for the value of the angle. Then check for all allowable values of the angle.    

                  5.    2 cos2 x – cos x = 1
    
                         2 cos2 x – cos x – 1 = 0

                         ( 2 cos x + 1 ) ( cos x – 1 ) = 0
               
                           2 cos x + 1 = 0                      
                           2 cos x =  – 1                         
                              cos  x  = – ½                                 
                             x =  120°  or  240°

                          cos x – 1  = 0
                          cos x  =  1
                                 x  = 0°   or  360°


                  6.   tan2 x – 3  =  0

                  7.   sin2 x – 1  = 0

                  8.   4 sin 2 x – 3  =  0 

                  9.    sin2 x  =  sin x 

                10.    cos2 x  =  ¼

                11.   3 tan2 x – 1  =  0

                12.   2 cos2 x  –  cos x  = 0 



LESSON  3


GRAPH OF TRIGONOMETRIC FUNCTIONS 

y = sin x 

             Data for  y = sin x
x
0
15
30
45
60
75
90
105
120
135
150
180
195
y
0.0
0.26
0.50
0.707
0.87
0.97
1.0
0.97
0.87
0.707
0.50
0.00
-0.26

210
225
240
255
270
285
300
315
330
345
360
-0.5
-0.707
-0.9
-0.97
-1.00
-1.0
-0.87
-0.707
-0.5
-0.3
0.0




y = cos x 

            Data for  y = cos x  
x
0
15
30
45
60
75
90
105
120
135
150
180
195
y
1.0
0.97
0.87
0.707
0.50
0.26
0.0
-0.3
-0.5
-0.707
-0.9
-1.0
-0.97

210
225
240
255
270
285
300
315
330
345
360
-0.9
-0.707
-0.5
-0.3
0.00
0.26
0.50
0.707
0.87
0.97
1.0


y =  tan x


LESSON  4

FUNCTIONS OF ANY ANGLE

     Sign of the functions in the four quadrants 


REDUCTION  FORMULAS

 I.   Any function of  [ (odd multiple of 90°)  ±  q ]  =  ± co-function of q

          a) second quadrant :  [ 90 + q ]  

          b) third quadrant :  [ 270 – q ]

          c) fourth quadrant : [ 270 + q ]


          Example 

                        1.  sin 120° =  sin [ 90° +  30° ]   =  cos 30°  =  0.8660254

                        2.  cos 215° =  cos [ 270° – 55° ]   = – sin 55°  =  – 0.81915204

                        3.  tan 302°  =  tan [ 270° + 32° ]  = – cot 32°  = – 1.600334  

                        4.  sin 150°     
                       
                       5.  tan 325°

 II.  Any function of [ (even multiple of 90° ± q) ]  =  ± same function of  q

             a) second quadrant :  [ 180  –  q ]  

             b) third quadrant :  [ 180  +  q ]

             c) fourth quadrant : [ 360 –  q ]


          Example 

           1.  sin 120° = sin [180°–  60° ]  =  sin 60° =  0.8660254

           2.  cos 215° = cos [180°+ 35° ]  = – cos 35° = – 0.81915204

           3.  tan 302°  = tan [360° – 58° ] = – tan 58° = – 1.600334  

           4.  sin 150°                 

           5.  tan 325°


  EXERCISES

  Reduce the following angles to its equivalent function of an acute angle.

               1.  sin 150°                              6.  cos 215°

               2.  tan 125°                              7.  sin 285°

               3.  cos 135°                             8.  cos 315°

               4.  tan 220°                              9.  tan 345°

               5.  cot  245°                          10.  cot 290°

           
LESSON  5

RADIAN – DEGREE CONVERSION

One radian is the measure of an angle which, if its vertex is placed at the center of a circle, intercepts on the circumference an arc equal in length to the radius.

                         one circumference  =  2 p radian

                                  360°  =  2 p

                                  180°  =  p

                                      1°  =  p /180 radian  = 0.017453 radian

                            1 radian  =  180/p  degrees

                            1 radian  =  57.29578°  =  57°  17’  45”


I.  To convert  radian to degree, multiply by  180/p.  ( 57.29578 )

    Example.  Convert  to degree.

                     1.  qp/3      
       
                     2.  q  = 2p/3       
              
                     3.  qp/6



            1. solution

                        q  =    p    x   180       =   60°
                                 3           p


            2.  solution

                        q  =   2 p    x   180      =   120° 
                                  3             p


            3.  solution

                        q  =   p     x   180    =   30°
                                 6            p 


  II.  To convert degree to radian, multiply by  p/180  or divide the angle by 57.29578.  

         Example.  Convert to radian.

            1.   q  =  150°                          

            2.  q  =  240°               

            3.  q  =  30°



            1.  solution

                        q  =  150° ( p/ 180 )  =  5p/6 radian


            2.  solution

                        q  =  240° ( p/180 )  =  4p/3 rad


            3.  solution

                        q  =  30 ( p/180 )  =  p/6 rad



 EXERCISES

  A.  Given are angles in radian, convert  to  degree.

            1.  q  = p/4                        5.   q  =  2.5 p             

            2.  q  = 2p / 3                    6.   q  =  2.5 rad           

            3.  q  =  1.5 rad                 7.   q  =  p/2    

            4.  q  =  0.75 p                  8.   q  =  4 rad 



  B.  Convert  to  radian. 

            1.  q  = 135°                      5.   q =  300°               

            2.  q  = 225°                      6.   q =  240°               

                      3.  q  =  270°                     7.   q  =  315°

            4.  q  =  140°                     8.   q  =  345°


LESSON 6


  ARC  AND  ANGLE  OF  A  CIRCLE

            S  =  r q 

     where
            
            S  ----  arc
            r   ----  radius of the circle
            q  ----  central angle in radian

Example
  1. A central angle of a circle with radius  of  15 cm  intercepts an arc of  30 cm.  
      What is the central angle in ( a ) radian and  (b ) degree ?

            Solution :

              S  =  r q

               q  =  S/r  =  30 cm / 15 cm  =  2  rad

               q  =  2 radian( 57.29578 )  =  114.59156°

  2. A central angle of  1.0472 radian subtends an arc of  5.236 cm.  Determine the radius 
      of the circle. What is the  angle in degree ?

 3.  A circle has a radius of 120 cm. Determine the central in degree subtended by an arc of 
      (a ) 60 cm,  (b ) 180 cm  and ( c ) 240 cm.

 4.  A circle has a radius of  80 cm. Determine the arc subtended by a central angle 
      of (a ) 60°,   (b ) 120°,  (c ) 150°  and  (d ) 240°.

 5.  A central angle of  2 radian subtends an arc of 40 cm. Determine the radius of the circle.
      What is the  angle in degree ?


SECTOR AND SEGMENT OF A CIRCLE

A sector is a portion of a circle bounded by two radii and their intercepted arc.

 A segment is a portion of a circle bounded by an arc and its chord.


Area of  sector, A  =  ½ r2 q ,     q is in radian
                  
Area of  segment,  A = ½ r2 ( q  –  sin q ) ,  q is in radian  and for sin qq  is in degree.

Example.
        1. Find the area of a sector of a circle whose radius is 30 cm and whose central 
            angle is 150°.

          Solution:

                        A = ½ r2 q  , 

                        q =  150 ( p/180 ) =  2.618

                        A = ½ (30 cm )2 ( 2.618 )

                        A =  1,178.1 cm2    
     
  2.  Find the area of a segment of a circle with radius of 40 cm if the central angle is 60°.

          Solution:

                        A = ½ r2 ( q  –  sin q )

                                q =  60 ( p/180 ) =  1.0472

                        A = ½ (40 cm )2 ( 1.0472 – sin 60° )

                        A =  144.94 cm2   


Exercises

1.  Find the area of a sector of a circle whose radius is 48 cm if the central angle is 240°.

2.  Find the area of a sector of a circle whose radius is 36 cm if the central angle 
     is ¾ p radian.

3.  Find the area of a sector of a circle whose radius is 60 cm if the central angle is 125°.

4.  Find the area of a sector of a circle whose radius is 50 cm if the central angle is 1.25 
     radian.

5.  Find the area of a segment of a circle with radius of 40 cm if the central angle is 120°.

6.  Find the area of a segment of a circle with radius of 30 cm if the central angle is ¾ p 
     radian.

7.  Find the area of a segment of a circle with radius of 40 cm if the central angle is 
    1.25 radian.

8.  Find the area of a segment of a circle with radius of 40 cm if the central angle is 210°.