Check the solution to "Solve the equation on the interval [0,2pi] sin⁡(x+π4)+sin⁡(x−π4)=1"

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glamrockqueen7

Answered question

2020-12-15

Solve the equation on the interval [0,2pi] $\sin (x + \frac{π}{4}) + \sin (x - \frac{π}{4}) = 1$

Answer & Explanation

wheezym

Skilled2020-12-16Added 103 answers

Use the sum and difference identities for sine:
$\sin (A \pm B) = \sin A \cos B \pm \cos A \sin B$
$(\sin x \frac{\cos}{π} 4 + \cos x \frac{\sin}{π} 4) + (\sin x \frac{\cos}{π} 4 + \cos x \frac{\sin}{π} 4) = 1$
$2 \sin x \frac{\cos}{π} 4 = 1$
$2 \sin x \cdot \frac{\sqrt{2}}{2} = 1$
$\sqrt{2} \sin x = 1$
$\sin x = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$
Sine is positive on QI and QII.
The reference angle is $\frac{π}{4}, since \frac{\sin}{π} 4 = \frac{\sqrt{2}}{2}$
The QI solution is:
$x = \frac{π}{4}$
The QII solution is:
$x = π - \frac{π}{4} = \frac{3 π}{4}$

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