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# Solving Trigonometric Equations Using Basic Algebra

## Substitute in potential solutions or solve using inverse trig functions.

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Practice Solving Trigonometric Equations Using Basic Algebra
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Solving Trigonometric Equations Using Algebra

As Agent Trigonometry, you are given this clue: 2sinx2=0\begin{align*}2 \sin x-\sqrt{2}=0\end{align*}. If 0x<2π\begin{align*}0 \le x < 2 \pi\end{align*}, what is/are the value(s) of x\begin{align*}x\end{align*}.

### Guidance

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

#### Example A

Verify that cscx2=0\begin{align*}\csc x-2=0\end{align*} when x=5π6\begin{align*}x=\frac{5 \pi}{6}\end{align*}.

Solution: Substitute in x=5π6\begin{align*}x=\frac{5 \pi}{6}\end{align*} to see if the equations holds true.

csc(5π6)21sin(5π6)2112222=0=0=0=0

This is a true statement, so x=5π6\begin{align*}x=\frac{5 \pi}{6}\end{align*} is a solution to the equation.

#### Example B

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

Solution: To solve this equation, we need to isolate cosx\begin{align*}\cos x\end{align*} and then use inverse to find the values of x\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.

2cosx+12cosxcosx=0=1=12

So, when is the cosx=12\begin{align*}\cos x=- \frac{1}{2}\end{align*}? Between 0x<2π,x=2π3\begin{align*}0 \le x< 2 \pi, x=\frac{2 \pi}{3}\end{align*} and 4π3\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 x=2π3±2πn\begin{align*}x=\frac{2 \pi}{3} \pm 2 \pi n\end{align*} and x=4π3±2πn\begin{align*}x=\frac{4 \pi}{3} \pm 2 \pi n\end{align*}, where n\begin{align*}n\end{align*} is any integer. You can check your answer graphically by graphing y=cosx\begin{align*}y=\cos x\end{align*} and y=12\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 5tan(x+2)1=0\begin{align*}5 \tan(x+2)-1=0\end{align*}, where 0x<2π\begin{align*}0 \le x < 2 \pi\end{align*}.

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

5tan(x+2)15tan(x+2)tan(x+2)=0=1=15

Using the tan1\begin{align*}\tan^{-1}\end{align*} button on your calculator, we get that tan1(15)=0.1974\begin{align*}\tan^{-1} \left(\frac{1}{5}\right)=0.1974\end{align*}. Therefore, we have:

x+2x=0.1974=1.8026

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

x=1.8026+π=1.3390=1.3390+π=4.4806

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

Concept Problem Revisit To solve this equation, we need to isolate sinx\begin{align*}\sin x\end{align*} and then use inverse to find the values of x\begin{align*}x\end{align*} when the equation is valid.

2sinx2=02sinxsinx=2=22

So now we need to find the values of x\begin{align*}x\end{align*} for which sinx=22\begin{align*}\sin x =\frac{\sqrt{2}}{2}\end{align*}. We know from the special triangles that this value of sine holds true for a 45\begin{align*}45^\circ\end{align*} angle, but is that the only value of x\begin{align*}x\end{align*} for which it is true?

We are told that 0x<2π\begin{align*}0 \le x < 2 \pi\end{align*}. Recall that the sine is positive in both the first and second quadrants, so sinx=22\begin{align*}\sin x =\frac{\sqrt{2}}{2}\end{align*} when x\begin{align*}x\end{align*} also is 135\begin{align*}135^\circ\end{align*}.

### Guided Practice

1. Determine if x=π3\begin{align*}x=\frac{\pi}{3}\end{align*} is a solution for 2sinx=3\begin{align*}2 \sin x=\sqrt{3}\end{align*}.

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

2. 3cos2x5=0\begin{align*}3 \cos^2x-5=0\end{align*}

3. 3sec(x1)+2=0\begin{align*}3 \sec(x-1)+2=0\end{align*}

1. 2sinπ3=3232=3\begin{align*}2 \sin \frac{\pi}{3}= \sqrt{3} \rightarrow 2 \cdot \frac{\sqrt{3}}{2}=\sqrt{3}\end{align*} Yes, x=π3\begin{align*}x=\frac{\pi}{3}\end{align*} is a solution.

2. Isolate the cos2x\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*}!

9cos2x59cos2xcos2xcosx=0=5=59=±53

The cosx=53\begin{align*}\cos x=\frac{\sqrt{5}}{3}\end{align*} at x=0.243\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 2π\begin{align*}2 \pi\end{align*}, x=2π0.243=6.037\begin{align*}x=2 \pi -0.243=6.037\end{align*} rad.

The cosx=53\begin{align*}\cos x=- \frac{\sqrt{5}}{3}\end{align*} at x=2.412\begin{align*}x=2.412\end{align*} rad, which is in the 2nd\begin{align*}2^{nd}\end{align*} quadrant. To find the other value where cosine is negative (the 3rd\begin{align*}3^{rd}\end{align*} quadrant), use the reference angle, 0.243, and add it to π\begin{align*}\pi\end{align*}. x=π+0.243=3.383\begin{align*}x= \pi+0.243=3.383\end{align*} rad.

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

3sec(x1)+23sec(x1)sec(x1)cos(x1)=0=2=23=32

At this point, we can stop. The range of the cosine function is from 1 to -1. 32\begin{align*}- \frac{3}{2}\end{align*} is outside of this range, so there is no solution to this equation.

### Explore More

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

1. x=7π6\begin{align*}x=- \frac{7 \pi}{6}\end{align*}
2. x=11π6\begin{align*}x=\frac{11 \pi}{6}\end{align*}
3. x=5π6\begin{align*}x=\frac{5 \pi}{6}\end{align*}

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

1. 1cosx=0\begin{align*}1- \cos x=0\end{align*}
2. 3tanx3=0\begin{align*}3 \tan x - \sqrt{3}=0\end{align*}
3. 4cosx=2cosx+1\begin{align*}4 \cos x=2 \cos x+1\end{align*}
4. 5sinx2=2sinx+4\begin{align*}5 \sin x-2=2 \sin x+4\end{align*}
5. secx4=secx\begin{align*}\sec x-4=- \sec x\end{align*}
6. tan2(x2)=3\begin{align*}\tan^2(x-2)=3\end{align*}

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

1. cosx=sinx\begin{align*}\cos x=\sin x\end{align*}
2. 3cscx=2\begin{align*}- \sqrt{3} \csc x=2\end{align*}
3. 6sin(x2)=14\begin{align*}6 \sin(x-2)=14\end{align*}
4. 7cosx4=1\begin{align*}7 \cos x -4=1\end{align*}
5. 5+4cot2x=17\begin{align*}5+4 \cot^2x=17\end{align*}
6. 2sin2x7=6\begin{align*}2 \sin^2x-7=-6\end{align*}