SOLUTION: solve the equation for x in [0 degrees, 360 degrees), give the answer to two decimal places : 5 sin 2x = 3

Question 635019: solve the equation for x in [0 degrees, 360 degrees), give the answer to two decimal places : 5 sin 2x = 3
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

Solving Trig equations like this usually involves three stages:

Use algebra and/or Trig properties to transform the equation into one or more equations of the form:
TrigFunction(expression) = number
Find the general solution for the equation(s) from stage 1. Trig. functions are periodic so there are usually an infinite number of solutions to these equations. The general solution expresses these infinite solutions.
Often these problems ask for a specific solution, like "Find the smallest positive solution to..." or "Find all solutions that are between p and q". To find specific solutions, you use the general solution.

Let's see this in action...

1. Transform.
With this equation the transformation stage is simple. Just divide both sides by 5:

2. Find the general solution.
We should recognize that 3/5 is not a special angle value for sin. So we will need our calculators. First we'll convert 3/5 to s decimal:
sin(2x) = 0.6
Then we will use our calculators to find the reference angle. Making sure your calculator is set to degree mode you should get:

Since this sin value is positive, it tells that the angle, 2x, is in the 1st or 2nd quadrants (where sin's are positive.) Combining the reference angle with the quadrants we get the equations:
2x = 36.86989765 + 360n (for the 1st quadrant)
2x = 180 - 36.86989765 + 360n (for the second quadrant)
with the second equation simplifying to:
2x = 143.13010235 + 360n
Now we solve for x by dividing both equations by 2:
x = 18.43494882 + 180n
x = 71.56505118 + 180n
Rounding to 2 decimal places (as the problem instructs) we get:
x = 18.43 + 180n
x = 71.57 + 180n
This is the general solution.
(Note: The "n" in these equations is a placeholder for any integer. To get specific x values, you replace "n" with an integer. Each integer value for n will result in an x value that is a solution to our equation. Some books/teachers use a different letter, like "k". The specific letter used is not important. What is important is that no matter what letter is used, it is a placeholder for an integer.)

3. Find the specific solution(s).
This is where we actually replace the n's in the general solution equations. We keep trying different n's until we feel we have found all the specific solutions. "[0 degrees, 360 degrees)" means . So we are looking for all x's between 0 and 360, including 0 but not including 360.
x = 18.43 + 180n
When n = 0, x = 18.43
When n = 1, x = 198.43
When n = 2 or larger, x is more than 360 (too big for )
When n is negative, x is negative (too small for )
x = 71.57 + 180n
When n = 0, x = 71.57
When n = 1, x = 251.57
When n = 2 or larger, x is more than 360 (too big for )
When n is negative, x is negative (too small for )
So all our specific solutions, in order from lowest to highest, are:
18.43, 71.57, 198.43 and 251,57.
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