Followers

Bidvertisers point

Thursday, June 25, 2015

MATH 11A ASSIGNMENT


MATH  11 A   8:00 – 10:00 MWF  
 assignment  to be submitted on June 29, 2015  


I.  Evaluate the value of  x  in the following absolute value expressions.

     1.    | 3x – 10 | = 4

     2.   |  12 – 5x | = 2

     3.  |  15 – 6x  | = 3

     4.  |  4x – 5  | = 3

     5.   | 10x –12 | = 8 

     6.   |  5x + 3 | =  7

     7.   |  8 + 3x |  =  2

     8.  |  7 – 3x |  =  1

 II. Do as indicated.

     1.  Find the sum of the expressions  2( 23x – 24y + 14 )  and   56y – 62x – 36. 

     2.  Find the sum of the expressions  17x2y + 12xy2 – 52w + 4   and  14w – 42xy2 – 49x2y – 38.   

     3.  Find the sum of the expressions ½( 24mp2 – 62xy + 46py – 8)  and  62xy – 46py – 28mp2 – 24).    

     4.  Subtract  the sum of   102xy – 56wx  + l9m2p  +34 and  24wx – 52xy – 21m2p – 22   from
           56m2p  + 36wx + 30xy – 12.    

     5.  Subtract  the sum of   12xy – 56px  + l9x2y  + 43  and  24px – 52xy – 14 x2y   – 22   from
           36x2y – 46px + 30xy – 62.    

     6.  Subtract  the sum of   23xy – 34px2  + 23xy2  + 63  and  24px2 – 62xy – 44xy2 – 38  from
           26xy2 – 63px2 + 50xy – 82.    

     7.  3(x2y)3(2px)2  

     8.  16(m2p)3/(2mx)2  

     9.  54(x-2y)3/(3x3y-3)3   

   10.  3axy(5axy)2  

   11.  3(4x – 5y) – 8x  + 12y

   12.  ½( 14m2 – 8xy2 + 12) – 4 + 2m2 + 10xy2    

     “Failure will never overtake you if your determination to succeed is strong enough”
                                                                                                                     - ifrain canhear ‘15


MATH  11 A   8:30 – 11:30 TTh  

assignment  to be submitted on June 30, 2015  


I.  Evaluate the value of  x  in the following absolute value expressions.

     1.    | 2x – 10 | = 6

     2.   |  12 – 4x | = 4

     3.  |  15 – 3x  | = 3

     4.  |  4x – 5  | = 7

     5.   | 8x –12 | = 4 

     6.   |  5x + 3 | =  2

     7.   |  8 + 4x |  =  4

     8.  |  7 – 2x |  =  3

II. Do as indicated.

     1.  Find the sum of the expressions  2( 32x – 44y + 24 )  and   26y – 42x – 16. 

     2.  Find the sum of the expressions  27x2y + 32xy2 – 22w + 8   and  24w – 32xy2 – 19x2y – 58.   

     3.  Find the sum of the expressions ½( 34mp2 – 42xy + 26py – 18) and 32xy – 16py – 38mp2 – 44).    

     4.  Subtract  the sum of   72xy – 66wx  + 29m2p  + 44 and  34wx – 32xy – 28m2p – 42   from
           36m2p  + 16wx + 45xy – 42.    

     5.  Subtract  the sum of   22xy – 46px  + 72x2y  + 62  and  44px – 62xy – 34 x2y   – 52   from
           56x2y – 66px + 20xy – 42.    

     6.  Subtract  the sum of   53xy – 36px2  + 43xy2  + 33  and  44px2 – 32xy – 24xy2 – 48  from
           66xy2 – 32px2 + 80xy – 42.    

     7.  3(2x2y)3(3px)2  

     8.  16(m2p)3/(4mx)2  

     9.  27(x-2y)3/(3x3y-3)3   

   10.  3axy(5axy)2  

    10.  ( 125 – x6 ) / ( x4 +  25 + 5x2)

    11.  ( 8m3 + 27p6 ) / ( 9p4 +  2m – 6mp2 )

    12.  ( 20m2 – 39m  + 18 ) / ( 5m – 6 )

     “Failure will never overtake you if your determination to succeed is strong enough”
                                                                                                                            - ifrain canhear ‘15


Saturday, June 20, 2015

AD RATES

Text Ads Prices
    Ad Product                   Cost                       Ad Length
   30 Day Image Ad        $ 20.00                      30 Days
     7 Day Image Ad          $ 4.00                        7 days
     3 Day Image Ad          $ 2.00                        3 Days
     1 Day Image Ad          $ 0.50                        1 Day
     Pay Per Click              $ 0.015                         N/A


 Image Ad Prices
   Ad Product                   Cost Image        Size           Ad Length
    30 Day Image Ad          $ 20.00           125x125        30 Days
      7 Day Image Ad          $ 6.00             125x125         7 days
      3 Day Image Ad            $ 2.00           125x125         3 Days
      1 Day Image Ad            $ 0.50           125x125         1 Day
       Pay Per Click               $ 0.015           125x125         N/A



Trigonometry : Lesson 1

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 + b2 )

                           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