Solving Trigonometric Equations, 7

Category: Trigonometry

"Published in Newark, California, USA"

Solve for the value of θ for

Solution:

Consider the given equation above

Convert the double angle function into its equivalent single angle function, we have

Take the square root on both sides of the equation, we have

If you will consider the positive sign:

Take the inverse tangent on both sides of the equation, we have

If you will consider the negative sign:

Take the inverse tangent on both sides of the equation, we have

Therefore, the answers are

where n is the number of revolutions.

Word Problem - Age Problem, 2

Category: Algebra

"Published in Newark, California, USA"

Mr. Manalang is 36 years old. His age is twice the sum of the ages of his two sons. How are old are his sons now, if the sum of the squares of their ages, 2 years ago, was 2 less than thrice Mr. Manalang's age then?

Solution:

The given word problem is about age problem where you need to solve for the age of a person or people. Let's analyze the given word problem above as follows: 

Let x = be the present age of Mr. Manalang's first son
      y = be the present age of Mr. Manalang's second son

If the first statement says, "Mr. Manalang is 36 years old. His age is twice the sum of the ages of his two sons.", then the working equation will be

If the second statement says, ".....if the sum of the squares of their ages, 2 years ago, was 2 less than thrice Mr. Manalang's age then?", then the working equation will be

   

If the first equation is 

Substitute the value of y from the first equation to the second equation, we have

Expand the above equation and solve for the value of x, we have

Equate each factor to zero, if

then the value of y will be equal to

And if,

then the value of y will be equal to

Therefore, the age of Mr. Manalang's sons are 10 years old and 8 years old. 
 

Derivative - Logarithmic Functions

Category: Differential Calculus, Algebra

"Published in Suisun City, California, USA"

Find the derivative for

Solution:

Consider the given equation above

If the logarithmic function has no base, then it is understood that the value of a base is 10. If the base of a logarithm is 10 or missing, then it is a common logarithm. 

Take the derivative of the above equation with respect to x using the formula for getting the derivative of a logarithmic function, we have

Take the derivative at the right side of an equation with respect to x using the formula for getting the derivative of rational functions, we have

  
Therefore,