What is the general solution for 3sin(x)sin(x)=5cos(x)-2 ?

The general solution for 3sin(x)sin(x)=5cos(x)-2 is x=0.79082…+2pin,x=2pi-0.79082…+2pin

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3sin(x)sin(x)=5cos(x)-2

3 sin (x) sin (x) = 5 cos (x) - 2

x = 0.79082 \dots + 2 πn, x = 2 π - 0.79082 \dots + 2 πn

Solution steps

3 sin (x) sin (x) = 5 cos (x) - 2

Subtract

5 cos (x) - 2

from both sides

3 sin^{2} (x) - 5 cos (x) + 2 = 0

Rewrite using trig identities

5 - 3 cos^{2} (x) - 5 cos (x) = 0

Solve by substitution

cos (x) = - \frac{5 + 85}{6}, cos (x) = \frac{85 - 5}{6}

cos (x) = - \frac{5 + 85}{6} :

No Solution

cos (x) = \frac{85 - 5}{6} : x = arccos (\frac{85 - 5}{6}) + 2 πn, x = 2 π - arccos (\frac{85 - 5}{6}) + 2 πn

Combine all the solutions

x = arccos (\frac{85 - 5}{6}) + 2 πn, x = 2 π - arccos (\frac{85 - 5}{6}) + 2 πn

Show solutions in decimal form

x = 0.79082 \dots + 2 πn, x = 2 π - 0.79082 \dots + 2 πn

\frac{cos ( x ) + 3 cos ( x )}{2 + 2} = 0

3 tan^{3} (x) - tan^{2} (x) - tan (x) - 1 = 0

cos (2 x + 60) = cos (x)

3 tan (x) - 3 cot (x) - 1 = 0

5 sin^{2} (x) + 6 cos (x) - 6 = 0

What is the general solution for 3sin(x)sin(x)=5cos(x)-2 ?
The general solution for 3sin(x)sin(x)=5cos(x)-2 is x=0.79082…+2pin,x=2pi-0.79082…+2pin