Solving Trigonometric Equations, 2

Category: Trigonometry

"Published in Newark, California, USA"

Solve for the unknown angle for

Solution:

The first that we have to do is to reduce the higher angles in order to simplify the given equation. Consider the given equation above

Apply the Sum and Product of Two Angles Formula in order to reduce the higher angles as follows

Equate each factor to zero, we have

for

and

where n = number of revolutions.

for

and

and

where n = number of revolutions.

Word Problem - Average Problem

Category: Algebra

"Published in Newark, California, USA"

Lilian's average in 3 quizzes is 80%. If she made 5% more in the first than in the second and 8% less in the third, what grades did she get in the three quizzes?

Solution:

The given item in a word problem is the average of the three quizzes of Lilian with conditions of her quizzes and we have to find the grades of her quizzes as follows

Let    x = the score of her 2nd quiz
        x + 5% = the score of her 1st quiz
        x - 8% = the score of her 3rd quiz

            Average of three quizzes = ⅓(1st Quiz + 2nd Quiz + 3rd Quiz)

                          80% = ⅓ (x + 5% + x + x - 8%)

                                   3x - 3% = 240%

                                           3x = 240% + 3%

                                           3x = 243%

                                             x = 81%

Therefore

Lilian's grade in 1st Quiz = x + 5% = 81% + 5% = 86%

Lilian's grade in 2nd Quiz = x = 81%

Lilian's grade in 3rd Quiz = x - 8% = 81% - 8% = 73%

First Order Linear Equation, 2

Category: Differential Equations, Trigonometry, Integral Calculus

"Published in Newark, California, USA"

Find the particular solution for

when

Solution:

Consider the given equation 

If 

then

If

then

Since

then the given equation is not an Exact Equation. We need to find the integrating factor first and then multiply it on both sides of an equation so that the given equation becomes an Exact Equation. 

Consider again the given equation

Arrange the above equation into its standard form as follows

where

and

The integrating factor is

The general solution for the above equation is 

If

then

Therefore, the particular solution for the above equation is 

We can consider the above equation as a final answer but if you want to further simplify the above equation, consider the Half Angle Formula as follows

Hence, the above equation becomes