Solve the equation. (find all solutions of the equation in the interval (0,2π). (7sin⁡(2x)+7cos⁡(2x))2=49

Question

Solve the equation. (find all solutions of the equation in the interval (0,2π).   (7sin⁡(2x)+7cos⁡(2x))2=49

Jayden-James Duffy · Accepted Answer

Given  (7sin⁡(2x)+7cos⁡(2x))2=49  solution  (7sin⁡(2x)+7cos⁡(2x))2=49  72(sin⁡(2x)+cos⁡(2x))2=49  (sin⁡(2x)+cos⁡(2x))2=1  sin2(2x)+cos2(2x)+2sin⁡(2x)cos⁡(2x)=1  1+2sin⁡(2x)cos⁡(2x)=1  2sin⁡(2x)cos⁡(2x)=0  sin⁡(2(2x))=0   (since, 2sin⁡xcos⁡x=sin⁡2x)  sin⁡4x=0  4x=sin−10  4x=nπ,n∈Z  x=π4,2π4,3π4,4π4,5π4,6π4,7π4,8π4,.....  Therefore the solution are,  x=π4,π2,3π4,π,5π4,3π2,7π4

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

Solve the equation. (find all solutions of the equation in the interval (0,2π). (7sin⁡(2x)+7cos⁡(2x))2=49

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