We are now going to combine our Algebra skills with our knowledge of Trig Identities and the Unit Circle and begin Solving Trigonometric Equations!
What makes these equations different than plain-old equations, is that they are conditional.
What’s a conditional equation?
An equation that is true for some, but not all values.
Don’t worry, this may sound tricky, but in all honesty, it’s very straightforward as long as you remember your Left Hand Trick (i.e., Reference Angles) from our amazing Unit Circle!
The main objective, as in solving any equation, is to isolate our trig function, using known algebraic methods such as addition, subtraction, factoring, etc., and then ask ourselves one very important question…
“What angle on the unit circle yields this length or coordinate?”
That’s it!
Truth be told, there are only two to three steps for Solving any Trigonometric Equation, and we are going to walk ourselves through this process with countless examples, just like the one you see below.
Solve the trig equation for all values of x
Also, we are going to look at an incredibly powerful trick for solving multiple angle or half-angle equations – it takes the sting right out of it, I promise!
Yes, as Purple Math states, you will need to put your thinking cap on, because sometimes your first instinct doesn’t get you any closer to a desired outcome, but taking a different approach will – don’t give up.
Moreover, as SOS Math points out, sometimes we will encounter solutions that are extraneous, or invalid, and that is why we must always check our answers.
The overall process is direct and uncomplicated, and with some practice, will become straightforward and clear.
Solving Trigonometric Equations – Video
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