Proving Trigonometric Identities, 5

Category: Trigonometry

"Published in Newark, California, USA"

Prove that 

Solution:

Consider the give equation above

Use the left side of the equation to prove the trigonometric identities because it is more complicated part, as follows

Use the sum and product formula, we have

Multiply the numerator and the denominator by sin x, we have

but

Therefore the above equation becomes

Therefore,

Theory - Polynomial Equations

Category: Algebra

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Form the equation of a polynomial if the roots are -3, 5, 6, -1, and -1. 

Solution:

The degree of a polynomial depends with the number of roots whether the roots are real numbers, irrational numbers, rational numbers, and complex numbers. In this case, if there are 5 roots in the given equation, then the degree of a polynomial must be 5 also. 

The number of terms of a polynomial is equal to the number of roots plus one. In this case, if there are 5 roots in the given equation, then the number of terms of a polynomial must be 6. 

If the roots are -3, 5, 6, -1, and -1, then the equation of a polynomial will be 

Eliminating Arbitrary Constant

Category: Differential Equations, Algebra

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Eliminate the arbitrary constant for

Solution:

Consider the given equation above and call this as equation #1, we have

Take the derivative with respect to x and call this as equation #2, we have

Since there are two arbitrary constants in the given equation, then we have to take the derivative of the given equation twice with respect to x. So, take the derivative of the above equation again with respect to x and call this as equation #3, we have

Using the equations #1 and #2:

Multiply equation #1 by 2 and multiply equation #2 by -1. Add the two equations in order to eliminate c₁ and call this as equation #4, we have

                                —————————————————

Using the equations #1 and #3:

Multiply equation #1 by 4 and multiply equation #3 by -1. Add the two equations in order to eliminate c₁ and call this as equation #5, we have

                                —————————————————

Equate equations #4 and #5, we have

Multiply both sides of the equation by 5, we have

Therefore,