Solve for x: 3*sin 5x = 4 + 3*cos 5x

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justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to solve 3*sin 5x = 4 + 3*cos 5x for x.

3*sin 5x = 4 + 3*cos 5x

=> 3(sin 5x - cos 5x) = 4

=> (sin 5x - cos 5x) = 4/3

=> (sqrt 2)(sin 5x*(1/sqrt 2) - cos 5x*(1/sqrt 2)) = 4/3

=> (sin 5x*cos 45 - cos 5x*sin 45) = 4/3*sqrt 2

=> sin (5x - 45) = (2*sqrt 2)/3

=> 5x - 45 = arc sin ((2*sqrt 2)/3)

=> 5x = 45 + 70.528

=> x = 115.5287/5

=> x = 23.10

The solution of the equation is 23.10+n*360 degrees.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll use the auxiliary angle method to solve the equation.

3 sin 5x - 3 cos 5x = 4

We'll divide entire equation by sqrt (3^2 + 3^2) = sqrt18.

(3/3sqrt2)*sin 5x - (3/3sqrt2)*cos 5x = 4/3sqrt2

We'll consider sin a = (3/3sqrt2) and cos a = (3/3sqrt2).

sin a*sin 5x - cos a*cos 5x = 4/3sqrt2

cos (a+5x) = - 4/3sqrt2

a + 5x = +/- arccos (- 4/3sqrt2) + 2k*pi

5x = +/- arccos (- 4/3sqrt2) + 2k*pi - a

x = +/- arccos (- 4/3sqrt2)/5 + 2k*pi/5 - a/5

The set of solutions of x is: {+/- arccos (- 4/3sqrt2)/5 + 2k*pi/5 - a/5}.

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