What is the general solution for sin^2(x)+3sin(x)-1=0 ?

The general solution for sin^2(x)+3sin(x)-1=0 is x=0.30760…+2pin,x=pi-0.30760…+2pin

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

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

x = 0.30760 \dots + 2 πn, x = π - 0.30760 \dots + 2 πn

Degrees

x = 17.62439 \dots^{\circ} + 36 0^{\circ} n, x = 162.37560 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

sin (x) = u

u^{2} + 3 u - 1 = 0

u^{2} + 3 u - 1 = 0 : u = \frac{- 3 + 13}{2}, u = \frac{- 3 - 13}{2}

u^{2} + 3 u - 1 = 0

Solve with the quadratic formula

u^{2} + 3 u - 1 = 0

Quadratic Equation Formula:

For

a = 1, b = 3, c = - 1

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 1 \cdot ( - 1 )}{2 \cdot 1}

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 1 \cdot ( - 1 )}{2 \cdot 1}

3^{2} - 4 \cdot 1 \cdot (- 1) = 13

3^{2} - 4 \cdot 1 \cdot (- 1)

Apply rule

- (- a) = a

= 3^{2} + 4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 3^{2} + 4

3^{2} = 9

= 9 + 4

Add the numbers:

9 + 4 = 13

= 13

u_{1, 2} = \frac{- 3 \pm 13}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- 3 + 13}{2 \cdot 1}, u_{2} = \frac{- 3 - 13}{2 \cdot 1}

u = \frac{- 3 + 13}{2 \cdot 1} : \frac{- 3 + 13}{2}

\frac{- 3 + 13}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 3 + 13}{2}

u = \frac{- 3 - 13}{2 \cdot 1} : \frac{- 3 - 13}{2}

\frac{- 3 - 13}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 3 - 13}{2}

The solutions to the quadratic equation are:

u = \frac{- 3 + 13}{2}, u = \frac{- 3 - 13}{2}

Substitute back

u = sin (x)

sin (x) = \frac{- 3 + 13}{2}, sin (x) = \frac{- 3 - 13}{2}

sin (x) = \frac{- 3 + 13}{2}, sin (x) = \frac{- 3 - 13}{2}

sin (x) = \frac{- 3 + 13}{2} : x = arcsin (\frac{- 3 + 13}{2}) + 2 πn, x = π - arcsin (\frac{- 3 + 13}{2}) + 2 πn

sin (x) = \frac{- 3 + 13}{2}

Apply trig inverse properties

sin (x) = \frac{- 3 + 13}{2}

General solutions for

sin (x) = \frac{- 3 + 13}{2}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{- 3 + 13}{2}) + 2 πn, x = π - arcsin (\frac{- 3 + 13}{2}) + 2 πn

x = arcsin (\frac{- 3 + 13}{2}) + 2 πn, x = π - arcsin (\frac{- 3 + 13}{2}) + 2 πn

sin (x) = \frac{- 3 - 13}{2} :

No Solution

sin (x) = \frac{- 3 - 13}{2}

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = arcsin (\frac{- 3 + 13}{2}) + 2 πn, x = π - arcsin (\frac{- 3 + 13}{2}) + 2 πn

Show solutions in decimal form

x = 0.30760 \dots + 2 πn, x = π - 0.30760 \dots + 2 πn

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

\frac{1 + ( 2 sin ( x ) )}{( cos ( x ) )} = 0

\frac{sin ^{2} ( a )}{2} = \frac{1 - cos ( a )}{2}

\frac{cot ^{2} ( x )}{2} = 3

sin (x - 3 0^{\circ}) = 0.7

What is the general solution for sin^2(x)+3sin(x)-1=0 ?
The general solution for sin^2(x)+3sin(x)-1=0 is x=0.30760…+2pin,x=pi-0.30760…+2pin