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Solving these equations usually starts with using algebra and/or Trig. identities to transform the equation so that you have one or more equations of the form:
trigfunction(expression) = number
It is not always easy to see how to make the desired transformation. When it is not obvious, then start by using Trig identities to reduce the number of different arguments or functions in the equation.
In this equation we have arguments of x and 2x. So one place to start is to use cos(2x) to change the argument of 2x to just x (matching the argument of sin). There are three variations of the cos(2x) formula:
Any of these can be used. I am going to use the third one because it has only sin in it. By using this variation not only will we get the arguments to match but we will also get all the functions in the equation to be sin's! Substituting the third variation in for cos(2x) in our equation we get:
Now let's simplify and see what we have. The "1" and the "-1" on the right cancel:
This is a quadratic equation for sin(x). So we start by getting one side to be zero. Adding to each side:
Then we factor:
Now we can use the Zero Product Property: or
The first equation is in the desired form. Subtracting 1 from both sides of the second equation: or
And then dividing by 2: or
Now both equations are in the desired form.
The next step is to write the general solution for each equation. The general solution expresses all the solutions to the equations. We'll start with:
We should recognize that 0 is a special angle value for sin. So we will need our calculators. We should know that sin is zero for 0, and for any other co-terminal angle. So our general solution for this is:
where "n" represents an integer. (NOTE: Some books/teachers use "k" here instead of "n". No matter what letter you use, "n" or "k" or any other letter, what is important is that the letter stands for an integer.) The "" is how we say "any any co-terminal angle".
Next we get the general solution for
We should recognize that -1/2 is also special angle value for sin. So we will need our calculators. We should know that sin of 1/2 has a reference angle of . And we should know that for sin to be negative the angle must terminate in the 3rd or 4th quadrants. So our general solution will be all angles which terminate in either the 3rd or 4th quadrant and which have a reference angle of : (or )
Simplifying: (or )
The complete general solution is all of these: (or )
Many, but not all, of these problems ask you to find a specific solution. For example: "Find the least positive solution to ..." or "Find all solutions to ... that are between zero and 360". When specific solutions are requested you use the general solution equation(s) and various integer values for "n" until you have found the requested solution(s).
In this case the problem does not ask for a specific solution. The general solution is the solution when no specific solution is requested.
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