What is the general solution for 4sin^2(x)-7sin(x)-2=0 ?

The general solution for 4sin^2(x)-7sin(x)-2=0 is x=-0.25268…+2pin,x=pi+0.25268…+2pin

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4sin^2(x)-7sin(x)-2=0

4 sin^{2} (x) - 7 sin (x) - 2 = 0

x = - 0.25268 \dots + 2 πn, x = π + 0.25268 \dots + 2 πn

Solution steps

4 sin^{2} (x) - 7 sin (x) - 2 = 0

Solve by substitution

sin (x) = 2, sin (x) = - \frac{1}{4}

sin (x) = 2 :

No Solution

sin (x) = - \frac{1}{4} : x = arcsin (- \frac{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 2 πn

Combine all the solutions

x = arcsin (- \frac{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 2 πn

Show solutions in decimal form

x = - 0.25268 \dots + 2 πn, x = π + 0.25268 \dots + 2 πn

cos (x) = - sin (2 x)

3 cot (x) - 3 = 0

cot (3 x) = \frac{3}{3}, 0 \leq x \leq 2 π

csc^{2} (θ) + 7 csc (θ) + 12 = 0

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

What is the general solution for 4sin^2(x)-7sin(x)-2=0 ?
The general solution for 4sin^2(x)-7sin(x)-2=0 is x=-0.25268…+2pin,x=pi+0.25268…+2pin