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Sunday, July 19, 2015

TRINALS TOP SCORERS


 CONGRATULATIONS



Top Scorers of TRINALS 2015

Given on July 15 - 16,  2015



PHYSICS E 102   -   Physics 2       Score

     1.  Largo,  Daisy Joy F   .                                         88
  2.  Intes, Emmanuel  C.                                           85
  3.  Salanatin,  Geo Austin G.                                   83
  4.  Macoy,  Leslie Joy  S.                                         81 





MATH 11A - COLLEGE ALGEBRA       Score

1.  Masinapoc, Jan Marie                                98
2.  Martizano, Luke Janrey                              96
3.  Dacuscus, Shenni                                      92
   Catubig, Christian                                      92
4.  Melocotones,  Robert                                90
   Limpioso, Princess Jhuner                        90
5.  Pancho,  Roxan                                         86
6.  Bauzon,  John Mark                                  84
   Claudio, Krista Eva                                    84
   Cabagua, Dexter                                       84
7.  Cabuguason,  Karen Kate                         82
   Blanco,  Justine  Mark                               82
8.  Glang,  Hannan                                         80
   Diamos,  Melvi                                           80




        PHYSICS 21 E                   Score

     1.  Puyod,  Nonezza                                            82
       Escote,  Michelle                                            82



MATH E 13 - PLANE TRIGONOMETRY Score

     1.  Jumamil,  Catherine H.                                         82
  2.  Presas,    Mark Bryan  C.                                     81






                                        Efren  F. Cadungog, Sr., MIM
                                        Class adviser


Friday, July 3, 2015

CITE AD

NDMC CITE advertisement edited by Michael John Destua. . . . .


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°.