SOLUTION: Solve the equation cosA/(1+sinA)+1+sinA/cosA=4, giving your answers for A in the interval 0<A<360

Question 816758: Solve the equation cosA/(1+sinA)+1+sinA/cosA=4, giving your answers for A in the interval 0
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
First of all, I assume that the second fraction's numerator is (1+sin(A)). If so, then please put multiple-term numerators (and denominators) in parentheses. If I am wrong then please re-post your question.

Here's a solution that may be faster than most. For reasons that will become clear soon, multiply the first fraction by (1-sin(A)):

Simplifying...

The factor of cos(A) in the numerator cancels one of the two factors of cos(A) in the denominator. (Do you see now why we multiplied by 1-sin(A)?)

The denominators are the same so we can add. The sin's cancel:

Multiplying both sides by cos(A):

Dividing by 4:

We should recognize that 1/2 is a special angle value for cos. It tells us that the reference angle is 60 degrees. Since the 1/2 is positive and since cos is positive in the first and fourth quadrants we should get general solution equations of:
A = 60 + 360n (for the first quadrant)
A = -60 + 360n (for the fourth quadrant)

Now we try various integers for n as we look for specific solutions within the given interval.
From A = 60 + 360n
if n = 0 then A = 60
if n = 1 (or larger) then A is too large for the interval
if n = -1 (or smaller) then A is too small for the interval
From A = -60 + 360n
if n = 0 (or smaller) then A is too small for the interval
if n = 1 then A = 300
if n = 2 (or larger) then A is too large for the interval

So the only solutions within the given interval are: 60 and 300.
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