Find all solutions of the equation in the interval {eq}\left[ 0,2\pi \right) {/eq}.
{eq}\sin \left( x+\dfrac{\pi }{3} \right)+\sin \left( x-\dfrac{\pi }{3} \right)=1 {/eq}
Question:
Find all solutions of the equation in the interval {eq}\left[ 0,2\pi \right) {/eq}.
{eq}\sin \left( x+\dfrac{\pi }{3} \right)+\sin \left( x-\dfrac{\pi }{3} \right)=1 {/eq}
Trigonometric Identities:
For any two angles {eq}A {/eq} and {eq}B {/eq}, we can convert the product of sine and cosine into the sum of sines as per the identity {eq}2\sin A\cos B = \sin \left( {A + B} \right) + \sin \left( {A - B} \right) {/eq}. The sine of the angle {eq}\dfrac{\pi }{3} {/eq} is {eq}\dfrac{{\sqrt 3 }}{2} {/eq} and its cosine value is {eq}\dfrac{1}{2} {/eq}.
Answer and Explanation: 1
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- Consider the equation {eq}\sin \left( {x + \dfrac{\pi }{3}} \right) + \sin \left( {x - \dfrac{\pi }{3}} \right) = 1 {/eq}.
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Chapter 22 / Lesson 1Trigonometric identities, or true trigonometric statements, have associated inverses which can be used to solve complex equations. Look at two example problems that require applying trigonometric identities, as well as their inverses, to be solved.