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

The general solution for 2sin(x)*tan(x)+5sin(x)=-2cos(x) is x=-0.46364…+pin,x=-1.10714…+pin

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2sin(x)*tan(x)+5sin(x)=-2cos(x)

2 sin (x) \cdot tan (x) + 5 sin (x) = - 2 cos (x)

x = - 0.46364 \dots + πn, x = - 1.10714 \dots + πn

Solution steps

2 sin (x) tan (x) + 5 sin (x) = - 2 cos (x)

Subtract

- 2 cos (x)

from both sides

2 sin (x) tan (x) + 5 sin (x) + 2 cos (x) = 0

Express with sin, cos

\frac{2 cos ^{2} ( x ) + 2 sin ^{2} ( x ) + 5 cos ( x ) sin ( x )}{cos ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Factor

2 cos^{2} (x) + 2 sin^{2} (x) + 5 cos (x) sin (x) : (2 sin (x) + cos (x)) (sin (x) + 2 cos (x))

(2 sin (x) + cos (x)) (sin (x) + 2 cos (x)) = 0

Solving each part separately

2 sin (x) + cos (x) = 0 or sin (x) + 2 cos (x) = 0

2 sin (x) + cos (x) = 0 : x = arctan (- \frac{1}{2}) + πn

sin (x) + 2 cos (x) = 0 : x = arctan (- 2) + πn

Combine all the solutions

x = arctan (- \frac{1}{2}) + πn, x = arctan (- 2) + πn

Show solutions in decimal form

x = - 0.46364 \dots + πn, x = - 1.10714 \dots + πn

2 cot (x) sin (x) + cot (x) = 0

2 sin (x) - (2 + 3) = - 3 csc (x)

7 cos^{2} (θ) + 6 sin (θ) - 10 = - 4

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

0 = - 2 sin (x) + cos (x)

What is the general solution for 2sin(x)*tan(x)+5sin(x)=-2cos(x) ?
The general solution for 2sin(x)*tan(x)+5sin(x)=-2cos(x) is x=-0.46364…+pin,x=-1.10714…+pin