Solving Trigonometry Problems

Solving Trigonometry Problems-82
Also note that sometimes we have to divide a sin by a cos to get a tan, as in one of the examples.And the last problem involves solving a trig inequality.

Also note that sometimes we have to divide a sin by a cos to get a tan, as in one of the examples.And the last problem involves solving a trig inequality.

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We can do this by adding \(2\pi k\) or \(\pi k\) where \(k\) is any integer (positive or negative).

Also note that a lot of times, when we get the solutions for tan, they are radians apart, so one set of solutions will the same as the other, and we can collapse into one solution and add \(\pi k\).

You won’t get the exact answers, but you can still compare to the exact answers you got above. This is because we could have fewer or more solutions in the Unit Circle, and thus for all real solutions when we add the \(2\pi k\) or \(\pi k\).

So when we solve these types of trig problems, we always want to solve for the General Solution first (even if we’re asked to get the solutions between solutions.

Also note that \(\) is written as \(\theta \), and we can put it in the graphing calculator as \(\boldsymbol\) or \(\boldsymbol \)., we have to first find the general solution of the equation, and then go back to the Unit Circle to see where the solutions are in the \(\left[ 0,2\pi \right)\) interval.

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We will learn how to do this Note that sometimes you may have to solve using degrees \(\left[ \right)\) instead of radians.\(\displaystyle \beginx \frac=\frac\,\,\,\,\,\,\,\,\,\,\,\,x \frac=\frac\\,\,\,x=-\frac\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x=\frac\end\) Note that (by looking at Unit Circle) this can be simplified to \(\displaystyle \) (Remember that you add \(\pi k\) instead of \(2\pi k\) for tan and cot).\(\displaystyle \begin\frac=\sin \theta \cos \theta \\cos \theta =4\theta \cos \theta \4\theta \cos \theta -\cos \theta =0\\cos \theta \left( \right)=0\\cos \theta \left( \right)\left( \right)=0\\cos \theta =0\text\,\,\text\,\,\sin \theta =\pm \frac\,\\\end\) Note that you can check these in a graphing calculator (radian mode) by putting the left-hand side of the equation into \(\) and the right-hand side into \(\) and get the intersection.We will mainly use the Unit Circle to find the exact solutions if we can, and we’ll start out by finding the solutions from \(\left[ 0,2\pi \right)\).We can also solve these using a Graphing Calculator, as we’ll see below.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 \(\displaystyle \theta \frac\).We learned how to factor Quadratic Equations in the Solving Quadratics by Factoring and Completing the Square section.Note that when we factor trig equations to find solutions, like we do with “regular” equations, we never just divide a factor out from each side.Note that when we multiply or divide to get the variable by itself, we have to do the same with the “\( 2\pi k\)” or “\( \pi k\)”.Again, watch for domain restrictions; answers that happen to fall on an asymptote for tan, cot, sec, or csc.


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