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sin (x + pi/3) + sin (x - pi/3) = 1 State all of the solutions on the interval (0,2). What is the...

Question:

{eq}sin (x + \frac{\pi}{3}) + sin (x - \frac{\pi}{3}) = 1 {/eq}

State all of the solutions on the interval {eq}[0,2) {/eq}. What is the strategy in determining which identities or technique to use in solving the equation?

Fundamental Trigonometric Equations

Let {eq}a {/eq} be a constant number in the interval {eq}[-1,1] {/eq}.

To solve the equation {eq}\sin x=a {/eq}, first, find a specific angle {eq}y {/eq} such that {eq}\sin y=a {/eq}. Then, all the solutions of {eq}\sin x=a {/eq} is given by

{eq}\displaystyle x=y+2k\pi\quad\text{or}\quad x=\pi-y+2k\pi\quad\text{where}\quad k=\ldots,-2,-1,0,1,2,\ldots. {/eq}

To solve the equation {eq}\cos x=a {/eq}, first, find a specific angle {eq}y {/eq} such that {eq}\cos y=a {/eq}. Then, all the solutions of {eq}\cos x=a {/eq} is given by

{eq}\displaystyle x=y+2k\pi\quad\text{or}\quad x=-y+2k\pi\quad\text{where}\quad k=\ldots,-2,-1,0,1,2,\ldots. {/eq}

Answer and Explanation: 1

Using the famous sum identities

{eq}\displaystyle \sin(a+b)=\sin a\cos b+\cos a\sin b,\quad \sin(a-b)=\sin a\cos b-\cos a\sin b, {/eq}

we have

{eq}\displaystyle \begin{align*} \sin\left(x+\frac{\pi}{3}\right)+\sin\left(x-\frac{\pi}{3}\right) &=\left(\sin x\cos \frac{\pi}{3}+\cos x\sin \frac{\pi}{3}\right)+\left(\sin x\cos \frac{\pi}{3}-\cos x\sin \frac{\pi}{3}\right)\\ &=2\sin x\cos \frac{\pi}{3}\\ &=\sin x. \end{align*} {/eq}

Therefore, the equation in the statement of the problem is equivalent to

{eq}\displaystyle \sin x=1. {/eq}

This equation has one solution in the interval {eq}[0,2) {/eq} and that is {eq}\boxed{x=\frac{\pi}{2}}. {/eq}


Learn more about this topic:

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What is Trigonometry? - Functions, Formulas & Applications

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Chapter 22 / Lesson 11
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Learn what trigonometry is and what trigonometric functions are. Understand the examples of how to use each function, as well as know the instances when it is useful to use trigonometry.


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