<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />

# 14.3: Solving Trigonometric Equations

Difficulty Level: At Grade Created by: CK-12

Objective

Here you’ll learn how to solve equations with trigonometric functions.

Review Queue

Solve the following equations.

1. \begin{align*}x^2-7x-18=0\end{align*}

2. \begin{align*}9x^4-16=0\end{align*}

3. \begin{align*}\sin x= \frac{\sqrt{2}}{2},0< x < \frac{\pi}{2}\end{align*}

## Using Algebra

Objective

Here you'll solve trig equations using algebra.

Guidance

In the previous concept, we verified trigonometric identities, which are true for every real value of \begin{align*}x\end{align*}. In this concept, we will solve trigonometric equations. An equation is only true for some values of \begin{align*}x\end{align*}.

Example A

Verify that \begin{align*}\csc x-2=0\end{align*} when \begin{align*}x=\frac{5 \pi}{6}\end{align*}.

Solution: Substitute in \begin{align*}x=\frac{5 \pi}{6}\end{align*} to see if the equations holds true.

This is a true statement, so \begin{align*}x=\frac{5 \pi}{6}\end{align*} is a solution to the equation.

Example B

Solve \begin{align*}2 \cos x+1=0\end{align*}.

Solution: To solve this equation, we need to isolate \begin{align*}\cos x\end{align*} and then use inverse to find the values of \begin{align*}x\end{align*} when the equation is valid. You already did this to find the zeros in the graphing concepts earlier in this chapter.

So, when is the \begin{align*}\cos x=- \frac{1}{2}\end{align*}? Between \begin{align*}0 \le x< 2 \pi, x=\frac{2 \pi}{3}\end{align*} and \begin{align*}\frac{4 \pi}{3}\end{align*}. But, the trig functions are periodic, so there are more solutions than just these two. You can write the general solutions as \begin{align*}x=\frac{2 \pi}{3} \pm 2 \pi n\end{align*} and \begin{align*}x=\frac{4 \pi}{3} \pm 2 \pi n\end{align*}, where \begin{align*}n\end{align*} is any integer. You can check your answer graphically by graphing \begin{align*}y=\cos x\end{align*} and \begin{align*}y=- \frac{1}{2}\end{align*} on the same set of axes. Where the two lines intersect are the solutions.

Example C

Solve \begin{align*}5 \tan(x+2)-1=0\end{align*}, where \begin{align*}0 \le x < 2 \pi\end{align*}.

Solution: In this example, we have an interval where we want to find \begin{align*}x\end{align*}. Therefore, at the end of the problem, we will need to add or subtract \begin{align*}\pi\end{align*}, the period of tangent, to find the correct solutions within our interval.

Using the \begin{align*}\tan^{-1}\end{align*} button on your calculator, we get that \begin{align*}\tan^{-1} \left(\frac{1}{5}\right)=0.1974\end{align*}. Therefore, we have:

This answer is not within our interval. To find the solutions in the interval, add \begin{align*}\pi\end{align*} a couple of times until we have found all of the solutions in \begin{align*}[0, 2 \pi]\end{align*}.

The two solutions are \begin{align*}x = 1.3390\end{align*} and 4.4806.

Guided Practice

1. Determine if \begin{align*}x=\frac{\pi}{3}\end{align*} is a solution for \begin{align*}2 \sin x=\sqrt{3}\end{align*}.

Solve the following trig equations in the interval \begin{align*}0 \le x< 2 \pi\end{align*}.

2. \begin{align*}3 \cos^2x-5=0\end{align*}

3. \begin{align*}3 \sec(x-1)+2=0\end{align*}

1. \begin{align*}2 \sin \frac{\pi}{3}= \sqrt{3} \rightarrow 2 \cdot \frac{\sqrt{3}}{2}=\sqrt{3}\end{align*} Yes, \begin{align*}x=\frac{\pi}{3}\end{align*} is a solution.

2. Isolate the \begin{align*}\cos^2x\end{align*} and then take the square root of both sides. Don’t forget about the \begin{align*}\pm\end{align*}!

The \begin{align*}\cos x=\frac{\sqrt{5}}{3}\end{align*} at \begin{align*}x=0.243\end{align*} rad (use your graphing calculator). To find the other value where cosine is positive, subtract 0.243 from \begin{align*}2 \pi\end{align*}, \begin{align*}x=2 \pi -0.243=6.037\end{align*} rad.

The \begin{align*}\cos x=- \frac{\sqrt{5}}{3}\end{align*} at \begin{align*}x=2.412\end{align*} rad, which is in the \begin{align*}2^{nd}\end{align*} quadrant. To find the other value where cosine is negative (the \begin{align*}3^{rd}\end{align*} quadrant), use the reference angle, 0.243, and add it to \begin{align*}\pi\end{align*}. \begin{align*}x= \pi+0.243=3.383\end{align*} rad.

3. Here, we will find the solution within the given range, \begin{align*}0 \le x< 2 \pi\end{align*}.

At this point, we can stop. The range of the cosine function is from 1 to -1. \begin{align*}- \frac{3}{2}\end{align*} is outside of this range, so there is no solution to this equation.

Problem Set

Determine if the following values for \begin{align*}x\end{align*}. are solutions to the equation \begin{align*}5+6 \csc x=17\end{align*}.

1. \begin{align*}x=- \frac{7 \pi}{6}\end{align*}
2. \begin{align*}x=\frac{11 \pi}{6}\end{align*}
3. \begin{align*}x=\frac{5 \pi}{6}\end{align*}

Solve the following trigonometric equations. If no solutions exist, write no solution.

1. \begin{align*}1- \cos x=0\end{align*}
2. \begin{align*}3 \tan x - \sqrt{3}=0\end{align*}
3. \begin{align*}4 \cos x=2 \cos x+1\end{align*}
4. \begin{align*}5 \sin x-2=2 \sin x+4\end{align*}
5. \begin{align*}\sec x-4=- \sec x\end{align*}
6. \begin{align*}\tan^2(x-2)=3\end{align*}

Sole the following trigonometric equations within the interval \begin{align*}0 \le x < 2 \pi\end{align*}. If no solutions exist, write no solution.

1. \begin{align*}\cos x=\sin x\end{align*}
2. \begin{align*}- \sqrt{3} \csc x=2\end{align*}
3. \begin{align*}6 \sin(x-2)=14\end{align*}
4. \begin{align*}7 \cos x -4=1\end{align*}
5. \begin{align*}5+4 \cot^2x=17\end{align*}
6. \begin{align*}2 \sin^2x-7=-6\end{align*}

Objective

Here you'll solve trig equations by factoring and the Quadratic Formula.

Guidance

Another way to solve a trig equation is to use factoring or the quadratic formula. Let’s look at a couple of examples.

Example A

Solve \begin{align*}\sin^2x-3 \sin x+2=0\end{align*}.

Solution: This sine equation looks a lot like the quadratic \begin{align*}x^2-3x+2=0\end{align*} which factors to be \begin{align*}(x-2)(x-1)=0\end{align*} and the solutions are \begin{align*}x = 2\end{align*} and 1. We can factor the trig equation in the exact same manner. Instead of just \begin{align*}x\end{align*}, we will have \begin{align*}\sin x\end{align*} in the factors.

There is no solution for \begin{align*}\sin x=2\end{align*} and \begin{align*}\sin x=1\end{align*} when \begin{align*}x= \frac{\pi}{2} \pm 2 \pi n\end{align*}.

Example B

Solve \begin{align*}1- \sin x=\sqrt{3} \cos x\end{align*} in the interval \begin{align*}0 \le x < 2 \pi\end{align*}.

Solution: To solve this equation, use the Pythagorean Identity \begin{align*}\sin^2x+\cos^2x=1\end{align*}. Solve for either cosine and substitute into the equation. \begin{align*}\cos x=\sqrt{1- \sin^2 x}\end{align*}

Solving each factor for \begin{align*}x\end{align*}, we get \begin{align*}\sin x=- \frac{1}{2} \rightarrow x=\frac{7 \pi}{6}\end{align*} and \begin{align*}\frac{11 \pi}{6}\end{align*} and \begin{align*}\sin x=1 \rightarrow x=\frac{\pi}{2}\end{align*}.

Example C

Solve \begin{align*}\tan^2x-5 \tan x-9=0\end{align*} in the interval \begin{align*}0 \le x < \pi\end{align*}.

Solution: This equation is not factorable so you have to use the Quadratic Formula.

\begin{align*}x \approx \tan^{-1} 6.41 \approx 1.416 \ \text{rad}\end{align*} and \begin{align*}x \approx \tan^{-1}-1.41 \approx -0.954 \ \text{rad}\end{align*}

The first answer is within the range, but the second is not. To adjust -0.954 to be within the range, we need to find the answer in the second quadrant, \begin{align*}\pi - 0.954=2.186 \ \text{rad}\end{align*}.

Guided Practice

Solve the following trig equations using any method in the interval \begin{align*}0 \le x < 2 \pi\end{align*}.

1. \begin{align*}\sin^2x \cos x=\cos x\end{align*}

2. \begin{align*}\sin^2x=2 \sin(-x)+1\end{align*}

3. \begin{align*}4 \cos^2x-2 \cos x-1=0\end{align*}

1. Put everything onto one side of the equation and factor out a cosine.

2. Recall that \begin{align*}\sin(-x)=- \sin x\end{align*} from the Negative Angle Identities.

The other solutions in the range are \begin{align*}x \approx 2 \pi - 0.6283 \approx 5.6549\end{align*} and \begin{align*}x \approx \pi + 1.2570 \approx 4.3982\end{align*}.

Problem Set

Solve the following trig equations using any method. Find all solutions in the interval \begin{align*}0 \le x < 2 \pi\end{align*}. Round any decimal answers to 4 decimal places.

1. \begin{align*}2 \cos^2x-\sin x -1=0\end{align*}
2. \begin{align*}4 \sin^2x+5 \sin x+1=0\end{align*}
3. \begin{align*}3 \tan^2x- \tan x=0\end{align*}
4. \begin{align*}2 \cos^2x+\cos(-x)-1=0\end{align*}
5. \begin{align*}1- \sin x=\sqrt{2} \cos x\end{align*}
6. \begin{align*}\sqrt{\sin x}=2 \sin x-1\end{align*}
7. \begin{align*}\sin^3x-\sin x=0\end{align*}
8. \begin{align*}\tan^2x-8 \tan x-7=0\end{align*}
9. \begin{align*}5 \cos^2x+3 \cos x-2=0\end{align*}
10. \begin{align*}\sin x- \sin x \cos^2x=1\end{align*}
11. \begin{align*}\cos^2x-3 \cos x-2=0\end{align*}
12. \begin{align*}\sin^2x \cos x=4 \cos x\end{align*}
13. \begin{align*}\cos x \csc^2x+2 \cos x=6 \cos x\end{align*}

Using your graphing calculator, graph the following equations and determine the points of intersection in the interval \begin{align*}0 \le x < 2 \pi\end{align*}.

1. .
1. .

## Date Created:

Apr 23, 2013

Jun 11, 2015
You can only attach files to section which belong to you
If you would like to associate files with this section, please make a copy first.