Proving Trigonometric Identities - Half Angles

Category: Trigonometry

"Published in Newark, California, USA"

Prove the trigonometric identity for 

Solution:

In proving the trigonometric identities, you need to examine the both sides of the equation very well. You have to choose the more complicated side of the equation and then simplify as much as you can until it is matched with the other side of the equation. In this case, the right side of the equation is more complicated. Let's start with simplifying the right side of the equation as follows

but

and hence

Therefore,

Finding Roots - Polynomial

Category: Algebra

"Published in Newark, California, USA"

Find the roots for the given polynomial:

x⁵ - 3x⁴ - x³ + 11x² - 12x + 4 = 0

Solution:

To solve for the roots of a given polynomial, we have to see the last term which is 4. Let's assign the factors of 4 which are 1, -1, 2, -2, 4, and -4.

First, let's check the number of positive and negative roots of a given polynomial using Descartes' Rule of Sign. Without changing the sign of x, the number of positive roots are 4 as shown below:

Next change x into -x and substitute to the given equation:

(-x)⁵ - 3(-x)⁴ - (-x)³ + 11(-x)² - 12(-x) + 4 = 0

-x⁵ - 3x⁴ + x³ + 11x² + 12x + 4 = 0

Since the highest degree of a given polynomial is 5, the total number of roots must be 5. There are 4 positive roots and 1 negative root of a given polynomial by Descartes' Rule of Sign. Let's see if the number of roots are correct by getting the factors using Synthetic Division, we have

x⁵ - 3x4 - x³ + 11x² - 12x + 4 = 0

If you will perform the Synthetic Division method, you have to arrange the polynomial in descending order first and then consider only their coefficient. In case that any of their middle term is missing, then you have to consider 0 as the coefficient of the missing term.

Continue to do the Synthetic Division as much as you can until you have one term left on the left side. You have to do this in trial and error using the factors of the last term so that the remainder is zero at the right side.

Therefore, the roots are 1, 1, 1, 2, and -2. 

Derivative - Increment Method

Category: Differential Calculus

"Published in Suisun City, California, USA"

Find the derivative of the given equation using the Increment Method:

Solution:

This is the original method in finding the derivative of any equations using the Increment Method. If you know already the List of Formulas to find a derivative of a certain equation, then you can use it. If you don't remember the formulas, then you have to use the Increment Method and it is a long process to do it. Well, for the Increment Method, you have to substitute x with x + Δx and substitute y with y + Δy, 

Subtract the given equation from the above equation, we have,

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

Simplify the above equation,

Divide both sides of the equation by Δx,

Therefore,

The above equation can be written as