Trigonometric Functions - 15, 75 Degrees

Category: Trigonometry

"Published in Suisun City, California, USA"

Without using a calculator, find the six trigonometric functions of 15º and 75º.

Solution:

Is it possible to find the six trigonometric functions for 15º and 75º without using a calculator? Well, we know the six trigonometric functions for the special angles such as 0º, 30º, 45º, 60º, and 90º. I am strongly advice that you must remember the trigonometric functions of special angles because you will use those later for proving of trigonometric identities, problem solving, and taking higher math subjects as well. Let's recall the six trigonometric functions of the special angles as follows

For 0º

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For 30º

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For 45º

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For 60º

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For 90º

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Therefore, using the Sum and Difference of Two Angles Formula, we have

Approximate Value - Square Root

Category: Differential Calculus, Arithmetic

"Published in Newark, California, USA"

Without using a calculator, find the approximate value of √101. 

Solution:

We know that the square root of 100 is 10. How do you get a square root of a number which is faster and easier than the traditional method without using a calculator? 

Well, let's consider this method using the principles of differentiation method. 

Let

then

If   x = 100
   dx = 101 - 100 = 1

then dy will be

Therefore

If you will use a calculator,√101 will be equal to 10.0498756211208902... which is close to the final answer.

Indeterminate Form - Infinity Raised Zero

Category: Differential Calculus, Algebra

"Published in Newark, California, USA"

Evaluate

Solution: 

Consider the given equation

Substitute the value of x to the above equation, we have

Since the answer is ∞⁰, then it is also another type of Indeterminate Form and it is not accepted as a final answer in Mathematics. We know that any number raised to zero power is always equal to one except for infinity that's why it is also an Indeterminate Form. In this type of Indeterminate Form, we cannot use the L'Hopital's Rule because the L'Hopital's Rule is applicable for the Indeterminate Forms like 0/0 and ∞/∞. Since the given equation is exponential equation, let's consider the following procedure

let 

Take natural logarithm on both sides of the equation

Substitute the value of x to the above equation, we have

  
Since the Indeterminate Form is 0∙∞, we have to rewrite the above equation as follows

Substitute the value of x to the above equation, we have

Since the Indeterminate Form is ∞/∞, then we can now use the L'Hopital's Rule as follows

Substitute the value of x to the above equation, we have

Since the Indeterminate Form is again ∞/∞, then we have to use the L'Hopital's Rule again as follows

Substitute the value of x to the above equation, we have

Since the Indeterminate Form is again ∞/∞, then we have to use the L'Hopital's Rule again as follows

Take inverse natural logarithm on both sides of the equation

Therefore,