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Let’s start out with solving fairly simple Trig Equations and getting the solutions from \(\left[ 0,2\pi \right)\), or \(\left[ \right)\).Here is the Unit Circle again so we can “pick off” the answers from it: Notice how sometimes we have to divide up the equation into two separate equations, like when the argument of the trig function is an expression, like \(\displaystyle \theta \frac\).As another example, for \(\cos \left( \frac \right)\), we’ll only get one solution instead of the normal two.
Solving trig equations is just finding the solutions of equations like we did with linear, quadratic, and radical equations, but using trig functions instead.
In doing this, we are probably “throwing away” valid solutions to the equation.
Here are some examples, both solving on the interval and over the reals; note one of the problems is using degrees instead of radians.
If we want \(\displaystyle \left( \right)\) for example, like in the The Inverse Trigonometric Functions section, we only pick the answers from Quadrants I and IV, so we get \(\displaystyle \frac\) only.
But if we are solving \(\displaystyle \sin \left( x \right)=\frac\) we get \(\displaystyle \frac\) and \(\displaystyle \frac\) in the interval \(\); there are no domain restrictions.