Question 235219: Solve sin^2 x - sin x - 2 = 0 for 0 < x < 360
Thanks a lot!
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! When you have problems like this, you need to be ready to use both Algebra and Trigonometry. You use Algebra to manipulate equations into a "friendlier" form, one you know how to solve. You use Trig.- To "extract" the argument of a function. For example, with
sin(2x+4) = 1
we use our knowledge that angles which are coterminal with 90 have a sin that is 1 to write:
2x+4 = 90 + 360n
The trig. function (sin) is now gone and the equation is now a relatively simple equation we can solve with Algebra. - To replace expressions, using Trig. identities. For example, we can take
sin(2x) - sin(x) = 0
and use the sin(2x) identity, sin(2x) = 2sin(x)cos(x), to rewrite this as
2sin(x)cos(x) - sin(x) = 0
which is more easily solved than the previous equation.
For your equation
sin^2 x - sin x - 2 = 0 for 0 < x < 360
we might consider using the Trig identity sin^2(x) = 1 - cos^2(x) to substitute for sin^2(x). But it doesn't seem to "improve" the equation. It doesn't look like the new equation would be easier to solve.
If we look at this from an algebraic viewpoint we will see a solvable quadratic equation. It might be easier to see this if we use a temporary variable. Let s = sin(x), Then our equation is
s^2 - s - 2 = 0
This is an easily solved quadratic equation. Just factor it:
(s-2)(s+1) = 0
Set the factors equal to zero and solve:
s-2 = 0 or s+1 = 0
s = 2 or s = -1
Of course we are not looking for "s". We are looking for x. So we substitute back for "s" ...
sin(x) = 2 or sin(x) = -1
(Eventually you will learn to "see" the quadratic nature of your original equation and factor and solve it without the use of a temporary variable.)
Now we can use our Trig. finish the solution:
1) sin(x) = 2. When is the sin of an angle 2? Answer: Never!! This means that there is no solution possible fron this equation.
2) sin(x) = -1. When is the sin of an angle -1? Answer: Any angle which is coterminal with 270. So the general solution is:
x = 270 + 360n
But we are asked only to find solutions between 0 and 360 and there is only one angle between 0 and 360 whose sin is -1: 270.
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