Proving Basic Trigonometric Identity

Category: Trigonometry, Plane Geometry

"Published in Newark, California, USA"

Prove that:

    
Solution:
 
Let's consider the right triangle
 

Photo by Math Principles in Everyday life

 

We know that 

and 

From Pythagorean Theorem,

divide both sides of the equation by c², we have

                                 or

Therefore,

                         

Deriving Double Angle Formula

Category: Trigonometry

"Published in Newark, California, USA"

Last October 9, 2012, we did the derivation of the Sum and Difference of Two Angles Formula. Right now, we will derive the formulas for Double Angle Formula from the formulas of the Sum and Difference of Two Angles Formula. 

Let's consider this one,

if θ = ϕ, then

Let's try this one,

if θ = ϕ, then

And finally, this one,

if θ = ϕ, then

Multiplying Three Digit Numbers

Category: Arithmetic

"Published in Newark, California, USA"

How do you multiply a three digit number with another three digit number without using a calculator in a shorter way and faster? I will show you a technique. Let's consider this one:

124 x 216

By usual method,
           1   2
           1  2  4
    x      2  1  6
 ---------------------
       1   7  4  4
       1  2  4
   2  4  8
----------------------
   2  6  7  8  4

Now, let's consider the other method which is Mayan Method. In this method, you will use the straight lines and their points of intersection.

For 124, draw 1 line, 2 parallel lines, and then 4 parallel lines that are all parallel to each other.

For 216, draw 2 parallel lines, 1 line, and then 6 parallel lines that are all parallel to each other.

Divide the figure into 5 sections. At each section, count the number of points which are the points of intersection.

Therefore, the answer is 26784.