Approximate Value - Sine, Cosine, Tangent

Category: Differential Calculus, Trigonometry

"Published in Newark, California, USA"

Find the approximate values of trigonometric functions for 31° without using a calculator or trigonometric tables.

Solution:

Is it possible to get the six trigonometric functions of 31° without using a calculator or trigonometric tables? Yes, we can. We can get the approximate values of trigonometric functions of 31° by approximation and error method. 

We know that 30° is a special angle. The trigonometric functions for 30° are the following

         Sin 30° = ¹/₂                         Csc 30° = 2

         Cos 30° = ^√3/₂                      Sec 30° = ²/_√3

         Tan 30° = ¹/_√3                      Cot 30° = √3

Let 

                                               

If    

Therefore

using a calculator

from trigonometric table

Let

Therefore

using a calculator

from trigonometric table

Let

Therefore

or

using a calculator 

from trigonometric table

Let

Therefore

or

using a calculator

from trigonometric table

Let

Therefore

or

using a calculator

from trigonometric table

Let

Therefore

or

using a calculator

from trigonometric table

Note: All angles are converted from degrees into radians because radians are unitless and all trigonometric values are unitless.

    

Solving 3 x 3 Determinants

Category: Algebra

"Published in Suisun City, California, USA"

Evaluate the determinant of a given matrix

Solution:

In expanding a 3 x 3 determinant into 2 x 2 determinants either by a row or by a column, you must remember that the signs of the members of a determinant are alternate either by row or by column as follows

Consider the given matrix

If you use the first row to expand the given determinant, the above determinant becomes

If you use the rest of the rows and columns of a given determinant, there are 5 other ways in expanding the determinant as follows

or

or

or

or

If you solve the five equations above, their value of determinant must be the same which is -44. 

There's another way in solving the 3 x 3 determinant without the expansion into 2 x 2 determinants by the rows or columns. Let's consider again the given matrix

Rewrite the first two columns of a given determinant as follows

There are three principal diagonals (top left to bottom right) and three secondary diagonals (bottom left to top right) in a given determinant as follows

          D = Sum of Principal Diagonals - Sum of Secondary Diagonals