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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
\n" ); document.write( "sin(2x+4) = 1
\n" ); document.write( "we use our knowledge that angles which are coterminal with 90 have a sin that is 1 to write:
\n" ); document.write( "2x+4 = 90 + 360n
\n" ); document.write( "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
\n" ); document.write( "sin(2x) - sin(x) = 0
\n" ); document.write( "and use the sin(2x) identity, sin(2x) = 2sin(x)cos(x), to rewrite this as
\n" ); document.write( "2sin(x)cos(x) - sin(x) = 0
\n" ); document.write( "which is more easily solved than the previous equation.
\n" ); document.write( "For your equation
\n" ); document.write( "sin^2 x - sin x - 2 = 0 for 0 < x < 360
\n" ); document.write( "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.
\n" ); document.write( "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
\n" ); document.write( "s^2 - s - 2 = 0
\n" ); document.write( "This is an easily solved quadratic equation. Just factor it:
\n" ); document.write( "(s-2)(s+1) = 0
\n" ); document.write( "Set the factors equal to zero and solve:
\n" ); document.write( "s-2 = 0 or s+1 = 0
\n" ); document.write( "s = 2 or s = -1
\n" ); document.write( "Of course we are not looking for \"s\". We are looking for x. So we substitute back for \"s\" ...
\n" ); document.write( "sin(x) = 2 or sin(x) = -1
\n" ); document.write( "(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.)
\n" ); document.write( "Now we can use our Trig. finish the solution:
\n" ); document.write( "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.
\n" ); document.write( "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:
\n" ); document.write( "x = 270 + 360n
\n" ); document.write( "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. \n" ); document.write( "