What is the general solution for 2sin(x)+csc(x)=3 ?

The general solution for 2sin(x)+csc(x)=3 is x= pi/6+2pin,x=(5pi)/6+2pin,x= pi/2+2pin

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

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

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{π}{2} + 2 πn

Solution steps

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

Subtract

3

from both sides

2 sin (x) + csc (x) - 3 = 0

Rewrite using trig identities

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

Solve by substitution

csc (x) = 2, csc (x) = 1

csc (x) = 2 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

csc (x) = 1 : x = \frac{π}{2} + 2 πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{π}{2} + 2 πn

2 cos (x) + 1 = 0, 0 \leq x \leq 2 π

arctan (x) = \frac{π}{4}

cos (2 x) (2 cos (x) + 1) = 0

8 cos^{2} (θ) = 4

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

What is the general solution for 2sin(x)+csc(x)=3 ?
The general solution for 2sin(x)+csc(x)=3 is x= pi/6+2pin,x=(5pi)/6+2pin,x= pi/2+2pin