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

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

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Popular >

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

Solution

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

Solution

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

Degrees

x = - 17.62439 \dots^{\circ} + 36 0^{\circ} n, x = 197.62439 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

sin (x) = u

u^{2} - 3 u = 1

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

u^{2} - 3 u = 1

Move

1

to the left side

u^{2} - 3 u = 1

Subtract

1

from both sides

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

Simplify

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

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

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 3)^{2} = 3^{2}

= 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}

Apply rule

- (- a) = a

= \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}

Apply rule

- (- a) = a

= \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} :

No Solution

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

- 1 \leq sin (x) \leq 1

No Solution

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

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

Popular Examples

tan(θ)= 12/19

tan (θ) = \frac{12}{19}

-cos(θ)=sin(θ)

- cos (θ) = sin (θ)

tan^2(θ)-tan(θ)=1

tan^{2} (θ) - tan (θ) = 1

sec(θ)= 8/5

sec (θ) = \frac{8}{5}

4tan^2(x)=sec^2(x)

4 tan^{2} (x) = sec^{2} (x)

Frequently Asked Questions (FAQ)

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