Followers

Bidvertisers point

Monday, March 27, 2017

TOP SCORERS FOR FINALS GIVEN LAST MARCH 22 - 25, 2017

CONGRATULATIONS

Top Scorers for FINALS

Given on March 22 – 24,  2017

PHYSICS E 22 – Modern Physics
         Name                              Score

1.   Geisler Cabayao                           96
2.  Korina  T.  Dayot                           94
       3.  Nathalie Gay C. Patricio               93
       4.  Carmel Therese Triumfante         92
       5.  Amerah  Ampatuan                       91
       6.  Kenneth  Gallano                          90
       7.  Aushe Mae Montebon                  86
       8.  Cristy Juliet  Reyes                       82  

 PHYS 21 Fundamentals of Physics
          Name                           Score

     1. Farouk Gumaan                              96
     2. Russel Ann  Completano                 95
     3. Bai Astrafiah Cassandra Abas        92
          Krizza Joy  Buckles                        92
     4. Baialmanisan  Dimasangkay           88


MATH 23/ MATH 201  Plane Trigonometry
          Name                          Score

     1.  Melanie  Talusan                          98
     2.  Angelo Florague                           84
     3.  Mhil Rose Zhine  Taytay               82
     4.  Benjie  Ibot                                   80


MATH 25  Analytic  Geometry
          Name                           Score

     1. Melanie  Talusan                           100
     2. Benjie  Ibot                                     92
     3. Ivy  Valenzuela                               90
         Johwena  Cuevas                          90
     4. Angelo Florague                            80  

MATH  11  College  Algebra
        Name                             Score

     1. Kevin Guiang                                   88
         Wahida Manial                                 88
     2. Winona Dojinog                               85  
     3. Kathleen Faye Dacanay                83

MATH 22C Automata & Language Theory
          Name                          Score

     1. Ritche Sal grande                           98
     2. Jay Harley Gadingan                      94
          Cristy Joy  Ybanez                         94
     3.  Charlotte Evangelio                       92
     4.  Joevin Niel Poquita                        91
     5.  Jeanifer  Aban                                86
     6.  Kenneth Jude Carrera                   83
     7.  Vicio Untal III                                  81

LANG 32 –  Foreign Language 2
         Name                             Score

1. Jane Mae Tampus                       100
           Al Suod Mascud                          100
           Mohamad Mustapha                   100
            Kimberly  Olmoguez                   100
            Wahida Manial                           100
            Kapia Sabpa                              100
       2. Benny  Chua                                 98
           Samraida  Mamay                        98
           Alchelou  Balogo                          98
       3. Adrian Camatac                            96
            Ervin  Ibot                                    96           
           Bainisan  Balad                            96
       4. Sheena Rose  Eduarte                84


                      Efren  F. Cadungog, Sr., MIM
                                 Class adviser 


Monday, February 13, 2017

Top Scorers for MIDTERM ( February 8 - 11, 2017 )


CONGRATULATIONS

Top Scorers for MIDTERM

Given on February 8 – 11,  2017

PHYSICS E 22 – Modern Physics
         Name                              Score

1.  Carmel Therese C.  Triumfante     95
Korina  T.  Dayot                            95
2.  Nathalie Gay C. Patricio                91
Geizler  O.  Cabayao                     91


 PHYS 21 Fundamentals of Physics                                 
          Name                              Score

     1.  Ana Marie Balanza                                83
     2.  Christine Joy Rocacorba                       81

PHYS 202  College Physics 2                                 
          Name                              Score

     1.  Bohiemean Flores                                92
     2.  Leovy Pearl Enoy                                 92


MATH 23/ MATH 201  Plane Trigonometry
          Name                             Score

     1.  Benjie Ibot                                            98
     2.  Melanie Talusan                                   94
     3.  Mhil Rose Zhine  Taytay                      90
     4.  Johwenna Cuevas                               89
     5.  Irev Marie Calawigan                           88
     6.  Bernovie Capre                                    85

MATH 25  Analytic  Geometry
          Name                            Score

     1.  Johwenna  Cuevas                              86
     2.  Benjie  Ibot                                           83
          Melanie  Talusan                                  83
     3.  Mhil Rose Zhine  Taytay                      80
          Shaina Kaye  Dadigan                         80

MATH  11  College  Algebra                                  
        Name                             Score

     1. Kevin Guiang                                      86
     2. Kathleen Faye  Dacanay                    84

MATH 22C Automata and Language Theory            

          Name                           Score

     1.  Retchie  Salmagrande                      80


LANG 32 –  Foreign Language 2                    
         Name                            Score

1. Alchelou Balogo                                87
          Sheena Rose Eduarte                      87
      2. Kimberly Olmoguez                          83
      3. Adrian Camatac                               80
          Kapia Sabpa                                    80
          Asnaira Milog                                   80



                  Efren  F. Cadungog, Sr., MIM
                            Class adviser 



Thursday, December 22, 2016

TOP SCORERS FOR TRINALS ON DECEMBER 14 - 15, 2016

CONGRATULATIONS

Top Scorers for TRINALS

Given on December 14 – 15,  2016

     MATH 25 Analytic Geometry
         Name                            Score

1.  Melanie P. Talusan                           84 

    MATH  201/ 23   Plane Trigonometry
         Name                            Score

      1. Angelo P. Florague                           84
2.  Benjie  E. Ibot                                   81
3. Rossianee  R.  Agnila                       80

    MATH  11   College Algebra
        Name                              Score

        1.  Florencio  Dinalo Jr.                       85
        2. Clemence Lloyd  S.  Uy                  80

  MATH  22 C  Automata & Language Theory
          Name                            Score

     1.  Joevin Niel  Poquita                          97
          Cristy Joy Ybañez                             97
     2.  Jay Harley  Gadingan                       90
     3.  Ritchie  P. Salmagrande                   84
     4.  Jose Rodolfo Fronda                        82
           Princess Pearl Nina Bayhonan        82 
     5.  Ivan Jay Bebit                                   80  

   LANGUAGE 102   
        Name                              Score

        1.  Asnaira U. Milog                             99
        2. Samraida H. Mamay                        93
        3. Joster Ann Tabian                            87
        4.  Kimberly  Olmoguez                       84
        5.  Ervin E.  Ibot                                   83




                       Efren  F. Cadungog, Sr., MIM
                        Class adviser 


Friday, December 9, 2016

Lesson 1 Trigonometry

TRIGONOMETRY

1. Trigonometry – is a branch of mathematics that deals with the relationship between the angles and sides of a triangle and the theory of the periodic functions connected with them.

2.  Triangle – a figure formed by three line segments joining three points that are not in the same plane.

3.  Right triangle – a triangle with at least one right angle.   

4.  Oblique triangle – any triangle which has no right angle. 
     Oblique triangles are classified according to sides and according to angles as :
        i) According to sides                   ii)  According to angles
           a) equilateral triangle                   a)  equiangular triangle 
           b) isosceles  triangle                    b)  acute triangle 
           c) scalene triangle                       c)  obtuse triangle 

5.  Congruent triangles – two triangles with the same shape and size. 

6.  Similar triangles – two triangles with the same shape but not necessarily of the same size.

7. Altitude of a triangle – the length of the line segment from any vertex of a triangle that is  perpendicular to the opposite side.

8.  Hypotenuse – the longest side of the triangle that is opposite to the right angle. 

9. Perimeter of a triangle – is the distance around the triangle and is equal to the sum of the three sides.

10. The area of a triangle is ½  of the base ( b ) times the altitude ( a ).  

                                                       A = ½ b a  

       If the three sides are given, the area is given by the formula
                                                              where  s =  ½ ( a + b + c ) 



RIGHT TRIANGLES  

                                                            Figure 1

11. Pythagorean theorem – in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

                                                   c2  =  a2  +  b2  

                                                   b2  =  c2 – a2  

                                                   a2  =  c2 – b2  


                 *  Proofs of Pythagorean theorem

                           ( as presented in the classroom discussion )


Examples :  Find the missing side, the perimeter and the area of the given right triangle. 

      1.  a =  12 cm ,  c =  13  cm                      7.   a  =  24 cm ,    c  =  40 cm 

      2.  b = 12 cm ,  a = 9 cm                           8.   b  =  36 cm ,    c  =  45 cm 

      3.  c = 35 cm,   b = 21 cm                         9.   a  =  33 cm ,    b  =  44 cm   

      4.  a = 36 cm,   b = 48 cm                        10.  b  =  48 cm ,     c  =  80 cm         

      5.  a = 81 cm,   c =  135 cm                     11.  a  =  63 cm ,     c  =  105 cm 
                            
      6.  c = 12 cm,   a =  6 cm                         12.  a  =  78  cm ,     b  =  104  cm   



    13.  Will a round glass table top, 2.5 m in diameter, fit through a doorway which is 2.13 m  
            high  and 0.92 m  wide ?

     14. A rectangle has a base of 12 cm. If the diagonal is 15 cm, determine the altitude of the rectangle.   

     15.  The area of a square is 128 cm2. Determine the length of its diagonal. 


     From Geometry :

                The sum of the measure of the three angles of a triangle is equal to  180°  

                                 From the figure above :

                                  a +  b  +  90°  =  180°   

                                  a +  b =  180° – 90°  

                                  a +  b = 90°  

  Hence, the sum of the measure of the acute angles of a right triangle is equal to 90°. 


   Example :  Find   a if  b  = 39° 28’.

                     Solution :    a  =  90° – 39° 28’  =  50°  32’  
                                         
      Exercises :  Find   a  or  b  if 

                 1.  a = 36° 24’  32’’                        6.  b = 56°  43’  27’’

                 2.  b =  68°  13’                             7.  a= 62° 54’  12’’

                 3.   a =  47° 19’                             8.   b = 73° 24’  32’’       

                 4.  a =  17°  21’  36”                      9.  a = 16° 48’ 

                 5.  b =  67°  34’  26”                     10.  b  = 22° 47’ 
            
    
PROPERTIES OF SIMILAR TRIANGLES 

Basic principle :
              The ratio of any two sides of a right triangle with an acute angle depends only on the size of the angle and not on the size of the triangle.
Memory aid
                        SOH                    CHO
                        CAH                    SHA
                        TOA                    CAO    


The six trigonometric ratios are called trigonometric functions. 

The value of each of the six trigonometric functions of an acute  angle  is determined  when  the acute angle is given.  Furthermore, it can be shown that, if the value of one of the six trigonometric functions of an acute angle is equal to the value of the same function of a second acute angle, the two acute angles are equal.

         Example 
                       1.    OA = 9 cm,  AX = 12 cm,   OX =  15 cm 
                              OB = 21 cm,  BY = 28 cm,   OY =  35 cm   
                              OC = 33 cm,  CZ = 44 cm,   OZ =  55 cm

                       2.   OA = 5 cm,  AX = 12 cm,   OX =  13 cm
                             OB = 10 cm,  BY = 24 cm,   OY =  26 cm   
                             OC = 20 cm,  CZ = 48 cm,   OZ =  52 cm

                       3.   OA = 6 cm,  AX = 8 cm,   OX =  10 cm
                             OB = 12 cm,  BY = 16 cm,   OY =  20 cm   
                             OC = 24 cm,  CZ = 32 cm,   OZ =  40 cm

  

FUNCTIONS OF ACUTE ANGLE

                        1.  Sin a                      4.  csc a
                        2.  cos a                      5.  sec a
                        3.  tan a                       6.  Cot a

Exercises

       1.  If sin a =  3/5, find cos a, tan a,  sec a,  csc a  and  cot a.

       2.  If tan a =  5/12, find cos  a, sin  a,  sec a,  csc  a  and  cot  a.

       3.  If cot a =  8/15, find cos  a, tan  a,  sec a,  csc  a  and  sin  a.

       4.  If cos a  =  12/13, find sin  a, tan  a,  sec  a,  csc  a  and  cot  a.

       5.  If csc a =  2, find cos a, tan a,  sec a,  sin a  and  cot a.

       6.  If sec  a =  5/4, find cos a, tan a,  sin  a,  csc  a  and  cot  a.

       7.  If cos  a =  4/5, find sin  a, tan  a,  sec  a,  csc  a  and  cot  a

       8.  If tan  a =  5/12, find cos  a, sin  a,  sec  a,  csc  a  and  cot  a.

       9.  Given a right triangle
                      Prove that    a)  sin a/cos a  =  tan a

                           b)  sin a cos a  =  ab /( a2 + b)

                           c)  ( sin  a ) 2  +  ( cos  a ) 2   =  1 
                                                                                                                                                 
                           d)  ( sec  a ) 2  –  ( tan  a ) 2   =  1

                           e)  ( csc  a ) 2  –  ( cot  a ) 2   =   1 


                                                                                                       
   10. In the figure, RS = 60 cm, cos a = 4/5, sin  = 5/13,  determine the length of the sides 
          QR, QS and  QP.                                                                                                
11.  Using the same figure, determine the unknowns if  sin a = 3/5, and cos O = 12/13 .

12.  Two guy wires are attached to a pole 12 m above the ground level. They make angles 
       of  43° and  72°  with the ground at points which are in a line with the base of the pole. 
       How long are the wires ?

13.  At low tide the angle of elevation to the top of  12 m tall tree from the water’s edge is 
      6° 30'. At high tide the angle of elevation to the top of the tree is  6°. How high does the          water  level rises during the high tide along the line perpendicular to the shore ?

14. Determine the height of a tower if from a window in a building 95.735 m away the angle 
    of depression to the base of the tower is 68° while the angle of elevation to the top is 62°.

15.  A light house casts a shadow of 9 m when the angle of elevation of the sun is 56°. How 
       high is the light house?

16. The angle of elevation to a balloon from a point on the ground is 32°.  After a vertical 
     ascent of  68 m the angle of depression from the balloon to the same point on the ground       is 47°. What are the original and present heights of the balloon ?

17. From a point 25 m from the base of a tower, a bird flies to its top in a straight path at an 
      angle of elevation of 75° 32’.  How long is the flight path of the bird ? What is the height 
     of the tower ?
                                      sin a   =  a/c                      cos b = a /c 

                                      cos a   =  b/c                     sin b = b /c 

                                      tan a   =  a /b                    cot b = a /b 

                                      sec a   = c /b                     csc b = c /b 

                                      csc a   =  c / a                   sec b = c / a 

                                      cot a  =  b / a                    tan b =  b / a 

                                    

                                     sin a  =  cos b  =  a / c 

                                     cos a  =  sin b  =  b / c 

                                     tan a  =  cot b  =  a / b 

                                     sec a  =  csc b  =  c / b 

                                     csc a  =  sec b  =  c / a 

                                     cot a  =  tan b  =  b / a


                        FUNCTIONS AND CO-FUNCTIONS 

                                    Sine  ---  cosine 

                                    Cosine ----  sine  

                                    Tangent --- cotangent  

                                    Cotangent ---  tangent 

                                    Secant ---  cosecant 

                                    Cosecant ---  secant  
              


                        RECIPROCAL  FUNCTIONS 

                                    sin a = 1/ csc a

                                    cos a  =  1/ sec a
                                                                      
                                    tan a  =  1/ cot a

                                    csc a  =  1/ sin a

                                    sec a  =  1/ cos a

                                    cot a  =  1/ tan a


Any function of a is equal to the co-function of the complement of a ( B ). 

      Examples :
                 1.  sin  30°  =   cos 60°  =  0.5 

                 2.  tan  15°  =  cot  75°  =  0.26794919 

                 3.   sec 36°  =  csc  54°  =  1.236067977

                 4.  cos  25°  =  sin  65°  =  0.906307787



SOLUTION OF RIGHT TRIANGLES                                                                      
                    Formulas :

                       1.  a2 +  b2  = c2

                        2.  a  +  b  = 90°                                             

                        3.  sin a  =  a / c  =  cos b                                       

                        4.  cos a  =  b / c  =  sin b

                        5.  tan a  =  a / b  =  cot b

                        6.  cot a  =  b / a  =  tan b

                        7.  sec a  =  c / b  =  csc b

                        8.  csc a  =  c / a  =  sec b

Example : 
     Find the acute angles of the triangle whose base is 15 cm and an altitude of 20 cm. 

                  From the figure above,  tan a =  20/15  =  1.333333333

                                                              a = Arc tan 1.3333333  =  53.13°       


                                                       tan b =  b/a  =  15/ 20  =  0.75

                                                            b  =  Arct tan 0.75  =  36.87°



   Exercises : Find the measure of the angle a and B of the following triangles using 
                     calculator.


            1.   a =  18,   b =  25                            6.  a =  21,  c =  40 

            2.  a = 40,  b = 60                                7.  b =  50,  c =  100

            3.  b = 17.321,  c = 20                         8.  a =  12,  c =  15  

            4.  a  =  12 ,  b =  24                           9.  a  =  15,  b = 28                                             

            5.  b  =  18,   c =  30                         10.  b  =  35,   c =  44 



SUGGESTIONS FOR SOLVING RIGHT TRIANGLES.

1. Make a preliminary sketch roughly to scale for the given data.

2.  To find any unknown part, use a formula which involves it but no other unknown. 

3.  Check the result by substituting it in any of the equations from  1 to 8.






ANGLE OF ELEVATION AND DEPRESSION

*  Angle of elevation – the angle formed by a horizontal ray  and the observer’s “line of sight” to any point above the horizontal.

*  Angle of depression – the angle formed by a horizontal ray and the observer’s “line of 
    sight” to  any point below the horizontal. 
  

Examples :

1.  A 120 m high tower casts a shadow of 60 m. Determine the angle of elevation of the sun.

2.  A light house is 14 m high. Determine the length of its shadow when angle of elevation of the sun is 52°.

3.  From an airplane flying at 2,135 m high above the ground, the angle of depression of a 
 landing field is 19° 32’. Determine the line of sight distance and the ground distance from 
 the plane to the field. (The ground distance is the distance of the plane from the field to the  point on the ground under the plane. ) 

4.  Determine the height of a tower if the angle of elevation of its top is  56° 35’ when seen from a point 206 m from the base of the tower.

5.  In flying upward  for  1,152 m along  a straight inclined path, an airplane rises  180 m. 
 Determine the climbing angle of the plane. ( Climbing angle is the angle of inclination of the plane from the horizontal ).

6.  From a ground distance of 1,100 m,  an airplane starts a straight glide for the edge of an airfield at a gliding angle of  16° from the horizontal. From what altitude did the glide starts ?



End of Lesson 1