SOLVING TRIGONOMETRIC EQUATIONS

Note: If you would like a review of trigonometry, click on trigonometry.

Problem 9.11d:        Solve for x in the equation

$$\displaystyle \frac{1-3\cos x}{\sin x}+ \displaystyle \frac{7\sin x}{1-\cos x}=3\csc x$$

Answer:    There are no solutions to this problem.

Solution:

Since denominators of fractions cannot equal zero, real numbers that cause the denominators to equal zero must be eliminated from the set of possible solutions. $\sin x\neq 0\rightarrow x\neq \pm n\pi ,\qquad$and         $\cos x\neq 1\rightarrow x\neq \pm 2n\pi .$ Therefore, before we even start to solve the problem, the set of real numbers in the set $\left\{ \pm n\pi \right\}$ must be excluded from the possible set of solutions.

The equation has two different trigonometric terms. Let's manipulate the equation so that we have an equation of like trigonometric terms.

First, let's multiple the second fraction by 1 in the form $$\displaystyle \frac{1+\cos x}{% 1+\cos x},$$ simplify and solve. The result will be an equation that is not equivalent to the original equation, but an equation where we can solve for x. With this type of manipulation, there may be extraneous answers. In other words, you may come up with answers for the new equation that are not solutions to the original equation. Therefore, check your answers with the original equation.

$$\begin{array}{rclll} && \\ \displaystyle \displaystyle \frac... ...ystyle \frac{5}{4}<-1\rightarrow \phi \\ && \\ && \end{array}$$

There is no real solutions.

If you forgot to make this observation, choose any number and try to check it by substituting it in the original equation. If, after the substitution, the left side of the original equation equals the right side of the original equation, the solution is valid. No real number will check.

You can also check the solutions graphically by graphing the function formed by subtracting the right side of the original equation from the left side of the original equation. The solutions of the original equation are the x-intercepts of this graph.

Algebraic Check:

Check solution any number, say 145.

Left Side:

$$\begin{array}{rclll} \displaystyle \frac{1-3\cos x}{\sin x}+\... ...1-\cos \left( 145\right) }\approx 24.6618339 \\ && \end{array}$$

Right Side:         $3\csc x\approx \displaystyle \displaystyle \frac{3}{\sin \left( 145\right) }\approx 6.41374886\bigskip$

Since the left side of the original equation does not equal the right side of the original equal when the number 145 is substituted for x, then 145 is not a solution.

Graphical Check:

Graph the equation $f(x)=\displaystyle \displaystyle \frac{1-3\cos x}{\sin x}+\displaystyle \displaystyle \frac{7\sin x}{1-\cos x}% -3\csc x.$ This function is formed by subtracting the right side of the original equation from the left side of the original equation. The x-intercepts are the solutions to the original equation.

Note that the graph never crosses the x-axis many times indicating no real number solutions.

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