Proving Trigonometric Identities - Higher Degree

Category: Trigonometry

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Prove the trigonometric identity for  

Solution:

Consider the given equation

We have to examine first the both sides of the equation for their complexity. Since the right side of the equation is complicated, then we will simplify the right side of the equation as follows

Therefore,

Algebraic Radicals

Category: Algebra

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Perform the indicated operations and simplify

Solution:

Consider the given equation

Get the Least Common Denominator (LCD) of the two grouped terms,

Get the reciprocal of the divisor and perform the multiplication

Since the denominator contains radicals, we need to rationalize the denominator by multiplying both sides of the fraction by its conjugate term, meaning the opposite sign of the other term. 

Therefore, the answer is

Converting Repeating Decimal - Fraction

Category: Arithmetic

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Express 0.454545454545454545454545454545....... into its equivalent fraction.

Solution:

The above decimal is a repeating decimal because the digit, 45 is repeating endlessly. It is usually happened if a fraction cannot be converted into basic fraction. Basic fraction is a fraction whose denominator is a multiple of 10. If the denominator is not a multiple of 10 but the multiples of its factors like 2 and 5 are given, then you can convert it into the multiples of 10. For example, if the denominator is 4, then you need to multiply both the numerator and the denominator by 25 so that the denominator will be 100. 

How about if the denominator is not a multiple of 10 and the factors are not the factors of 10? Well, there's no way to convert it into the multiples of 10. For example if the denominator is 35, then the factors are 5 and 7. Since 7 is not a factor of 10, then there's no way to convert the denominator into the multiples of 10. If you will do the manual division by dividing the numerator by denominator, you will notice that there's no end in division because the remainder is always the same and the digits of the quotient are repeating endlessly that's why the given decimal is called a repeating decimal.

Let's go back to the problem. How do you convert the repeating decimal into a fraction? Well, there's a way to convert the repeating decimal into a fraction. Here's the procedure and solution

Let     x = 0.454545454545454545454545454545.......... 

   100 x = 45.45454545454545454545454545454545......

Subtract x from 100 x, we have

   100 x = 45.4545454545454545454545454545454545......
        - x = -0.454545454545454545454545454545454......
   -------------------------------------------------------------------------------
     99 x = 45

Divide both sides of the equation by 99

As a rule in Mathematics that all fractions must be simplified into a lowest term. Factor 45 and 99 as follows

Their Greatest Common Factor (GCF) is 9. Cancel 9 on both sides of the fraction. Therefore, the answer is