Proving Trigonometric Identities - Inverse Function

Category: Trigonometry

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Prove the trigonometric identity for

Solution:

The first thing that you have to do is to examine the both sides of the equation and look for the more complicated side. In this case, the left side of the equation is more complicated. Let's simplify the left side of the equation as follows

The left side of the equation is the sum of two angles of tangent function. Let's apply the sum of two angles formula for tangent function as follows

Since each trigonometric functions at each term of the equation are inverse functions to each other, then we can cancel the tangent and inverse tangent functions at each term as follows

Therefore, 

Multiplying Both Two - Three Digit Numbers

Category: Arithmetic

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How do you multiply these numbers in a shorter way without using a calculator: 
                        
                     66 x 64

Solution:

Well, we have the technique for that. What is the next number after 6? 7, isn't it? Therefore, 6 x 7 = 42. 42 will be the first two digits of the number.

                     4 2 _ _

How do you get the next two digits? Well, multiply both the one's digits in order to get the next two digits. Therefore, 6 x 4 = 24. 24 will be the next two digits of the number. The answer is

                     4 2 2 4

Note: There are two conditions in order for you to use this technique:

1. The ten's digits must be the same. In this case, both are starting with 6. 

2. The sum of the two one's digits must be equal to ten. In this case, 6 + 4 = 10.

Now, you know this technique. Let's try another one:

                     21 x 29

Solution:

The first two digits will be 2 x 3 = 6. Since the product is less than 10, the first digit will be 6. The next two digits will be 1 x 9 = 9. Since the product is less than 10, the next two digits must be written as 09. The answer is

                     6 0 9

Can we do this for both three digit numbers? Yes, we can do this also for both three digit numbers as long as the two conditions are followed. Let's try this one:

                   112 x 118

Solution:

As you notice that both numbers are starting with 11. The first digits will be 11 x 12 = 132. To multiply any two digit number by 11, place the first digit as the first digit of the product, place the last digit as the last digit of the product, and the middle digit of the product will be the sum of the two digits. In this case, the middle digit is 1 + 2 = 3. If the sum of the digits is 10 or more, use the ten's digit as a carry over to the first digit. 

The next two digits will be 2 x 8 = 16. Therefore, the answer is 

                    1 3 2 1 6 

Integration - Trigonometric Functions

Category: Integral Calculus, Trigonometry

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Integrate

Solution:

Integrating the trigonometric functions is not the easy one because you will use the trigonometric identities often. You must remember or memorize all the trigonometric identities and formulas so that you can integrate the trigonometric functions very well. Let's start for the given problem. First, group the Sin x and Cos 2x in the given equation

Apply the Sum and Product of Two Angles Formula for the grouped term

At the first term of the above equation, we cannot integrate the trigonometric function immediately because the trigonometric function has exponent. If the du of the first term is present, then we can integrate it by integration of power. We can convert the trigonometric function of the first term into a single exponent by using the Half Angle Formula.

At the second term of the above equation, the product of two trigonometric functions have different angles. We have to use the Sum and Product of Two Angles Formula again.