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

glamrockqueen7

glamrockqueen7

Answered question

2020-12-15

Solve the equation on the interval [0,2pi] sin(x+π4)+sin(xπ4)=1

Answer & Explanation

wheezym

wheezym

Skilled2020-12-16Added 103 answers

Use the sum and difference identities for sine:
sin(A±B)=sinAcosB±cosAsinB
(sinxcosπ4+cosxsinπ4)+(sinxcosπ4+cosxsinπ4)=1
2sinxcosπ4=1
2sinx22=1
2sinx=1
sinx=12=22
Sine is positive on QI and QII.
The reference angle is π4, since sinπ4=22
The QI solution is:
x=π4
The QII solution is:
x=ππ4=3π4

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