NATURAL SINES AND NATURAL COSINES

HOW TO USE NATURAL SINES AND NATURAL COSINES CALCULATIONS WITH ILLUSTRATIONS

 

   Most important trigonometry formulae :

          sin(90º - θ) = cos(θ)

          sin(90º + θ) = cos(θ)

          cos(90º - θ) = sin(θ)

          cos(90º + θ) = -sin(θ)

          sin(180º - θ) = sin(θ)

          sin(180º + θ) = -sin(θ)

          cos(180º - θ) = -cos(θ)

          cos(180º + θ) = -cos(θ)

           where,θ is angle

      1º = 60´ i.e one degree equal sixty minute. 

 ( ºsymbol of degree  and  ´symbol of minute )

           0.1º =  6´  → ( 0.1º x 60´ =  6´ )

           0.2º = 12´ → ( 0.2º x 60´ =  12´ )

           0.3º = 18´ → ( 0.3º x 60´ =  18´ )

           0.4º =  24´→ ( 0.4º x 60´ =  24´ )

           0.5º = 30´ → ( 0.5º x 60´ =  30´ )

           0.6º = 36´ → ( 0.6º x 60´ =  36´ )

           0.7º = 42´ →  ( 0.7º x 60´ =  42´ )

           0.8º = 48´ →  ( 0.8º x 60´ =  48´ )

           0.9º = 54´ →  ( 0.9º x 60´ =  54´ )

           

      Method of using the table of sines and cosines: this table is also known as the table of natural sines and natural cosines.At intervals of 60', we can find the values of sines and cosines of angles ranging from 0° to 89°.

         The table of natural sines and natural cosines is essentially divided into the following sections, as shown by the following tables. 

        

       (i) the angles in the table's extreme left vertical column are from 0° to 89° at intervals of 1°. 

 

        (ii) at intervals of 6' , the horizontal row at the top of the chart is from 0' to 54'.

        (iii) at intervals of 1', the angles in the horizontal row at the top of the chart are from 1' to 5'.

               This portion of the chart is referred to as the mean difference column.

* Note: 

           (i) we obtain the sine or cosine value of a given angle from the chart, which is expressed in four decimal places. 

           (ii) we know that the sine of any given angle is equal to that of the cosine of its complementary angle, i.e., sin(θ) =cos(90º - θ) . The table is also drawn in such a way that we can determine the sin and cosine values of any given angle between 0° and 89°.

             (iii) Cosine this portion of the chart is referred to as the subtract mean difference column.

Solved examples using the table of natural sines : 

         1. Find the value of sin 44°. 

Solution:   Find the value of sin 44° by the using the table of natural sines we need to go through the extreme left vertical column 0° to 89° and move downwards till we reach the angle 44°.               

               Then we move horizontally to the right at the top of the column headed by 0' (or 0.0°) and read the figure 0.6947, which is the require value of sin 44°. Therefore sin 44° = 0.6947

          2. Find the value of sin 62°24’

Solution:  Find the value of sin 62°24’ by the using the table of natural sines we need to go through the extreme left vertical column 0° to 89° and move downwards till we reach the angle 62°. 

           Then we move horizontally to the right at the top of the column headed by 24' (or 0.4°) and read the figure 0.8862, which is the require value of sin 62°24’ .

        Therefore sin 62°24’  = 0.8862

            3. Using the trigonometric table, find the value of sin 62°28'

Solution: To find the value of sin 62°28' by the using the trigonometric table of natural sines we need to first find the value of sin 62°24'.

           To find the value of sin 62°24’ by the using the table of natural sines we need to go through the extreme left vertical column 0° to 89° and move downwards till we reach the angle 62°. 

           Then we move horizontally to the right at the top of the column headed by 24' (Or 0.4°)and read the figure 0.8862, which is the require value of sin 62°24'. 

       Therefore, sin 62°24' = 0.8862

         Now we move further right along the horizontal line of angle 62° to the column headed by 4' of mean difference and read the figure 5 there; this figure of the table does not contain decimal sign. In fact, 5 implies 0.0005. Now we know that when the value of an angle increases from 0° to 89°, its sine value increases continually from 0 to 1. Therefore, to find the value of sin 62°24' we need to add the value corresponding to 4’ with the value of sin 62°24'.                             

   Therefore, sin 62°28' = sin ( 62°24' + 4') = 0.8862 + 0.0005 = 0.8867 

             4.Use tables to find sin77°78' 

Solution: Using the trigonometric table of natural sine  sin77°78' = sin(77°60' + 28')                             But 60' = 1°

Therefore  sin77°78' = sin(78° + 28' ) = sin(78°+ 28') = sin(78°24' + 4')= 0.9796 + 0.0002 = 0.9798  

 sin77°78' = 0.9798     ... Answer

          5. Find the angle θ, sin θ = 0.9298

solution : sin θ = 0.9298 

Therefore   θ = sin⁻¹ 0.9298 = 68° 24′ because in the table, the value 0.9298 corresponds to the column of 24′ in the row of 68°.

   θ = sin⁻¹ 0.9298 = 68° 24′ .... Answer

Solved examples using the table of natural cosines : 

   1. Find the value of cos 63°28' ,Using the trigonometric table. 

solution:  The value of cos 63°28' = cos ( 63°24' + 4' ) = ( 0.4478 - 0.0010 mean difference subtract ) = 0.4468 Therefore  cos 63°28' = 0.4468  ... Ans

   2. Find the value of cos 25°.  

 Solution: Find the value of cos 25° by the using the table of natural cosines.  The extreme left column consisting of 0° to 89° and move downwards till we reach the angle 25°.    

         We then move horizontally to the right at the top of the column headed by 0' (or 0.0°) and say the figure 0.9063, which is the demand value of 25°.      

Therefore cos 25° = 0.9063 ... Answer

    3. Find the value of cos 28.9°.

  Solution: The value of cos 28.9°, using the natural cosine table cos 28.9° = cos (28° + 0.9° ) = cos 28° 54' = 0.8755 Therefore , cos 28.9°= 0.8755.... Ans