What is the general solution for sin(x)+cot(x)cos(x)=sqrt(3) ?

The general solution for sin(x)+cot(x)cos(x)=sqrt(3) is x=0.61547…+2pin,x=pi-0.61547…+2pin

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sin(x)+cot(x)cos(x)=sqrt(3)

sin (x) + cot (x) cos (x) = 3

x = 0.61547 \dots + 2 πn, x = π - 0.61547 \dots + 2 πn

Solution steps

sin (x) + cot (x) cos (x) = 3

Subtract

3

from both sides

sin (x) + cot (x) cos (x) - 3 = 0

Express with sin, cos

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

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

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

Rewrite using trig identities

- 3 sin (x) + 1 = 0

Move

1

to the right side

- 3 sin (x) = - 1

Divide both sides by

- 3

sin (x) = \frac{3}{3}

Apply trig inverse properties

x = arcsin (\frac{3}{3}) + 2 πn, x = π - arcsin (\frac{3}{3}) + 2 πn

Show solutions in decimal form

x = 0.61547 \dots + 2 πn, x = π - 0.61547 \dots + 2 πn

4 sin^{4} (x) + 3 sin^{2} (x) - 1 = 0

3 tan (θ) - 3 = 0

cot (θ) = \frac{1}{cot ( θ )}

(tan (x))^{5} - 9 tan (x) = 0

2 cot^{2} (x) + 2 csc^{2} (x) = 1 + 4 csc (x)

What is the general solution for sin(x)+cot(x)cos(x)=sqrt(3) ?
The general solution for sin(x)+cot(x)cos(x)=sqrt(3) is x=0.61547…+2pin,x=pi-0.61547…+2pin