What is the general solution for 5sin^2(x)=3sin(x)+2 ?

The general solution for 5sin^2(x)=3sin(x)+2 is x= pi/2+2pin,x=-0.41151…+2pin,x=pi+0.41151…+2pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

5sin^2(x)=3sin(x)+2

Solution

5 sin^{2} (x) = 3 sin (x) + 2

Solution

x = \frac{π}{2} + 2 πn, x = - 0.41151 \dots + 2 πn, x = π + 0.41151 \dots + 2 πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = - 23.57817 \dots^{\circ} + 36 0^{\circ} n, x = 203.57817 \dots^{\circ} + 36 0^{\circ} n

Solution steps

5 sin^{2} (x) = 3 sin (x) + 2

Solve by substitution

5 sin^{2} (x) = 3 sin (x) + 2

Let:

sin (x) = u

5 u^{2} = 3 u + 2

5 u^{2} = 3 u + 2 : u = 1, u = - \frac{2}{5}

5 u^{2} = 3 u + 2

Move

2

to the left side

5 u^{2} = 3 u + 2

Subtract

2

from both sides

5 u^{2} - 2 = 3 u + 2 - 2

Simplify

5 u^{2} - 2 = 3 u

5 u^{2} - 2 = 3 u

Move

3 u

to the left side

5 u^{2} - 2 = 3 u

Subtract

3 u

from both sides

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

Simplify

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

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

Write in the standard form

a x^{2} + b x + c = 0

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = 5, b = - 3, c = - 2

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 5 ( - 2 )}{2 \cdot 5}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 5 ( - 2 )}{2 \cdot 5}

(- 3)^{2} - 4 \cdot 5 (- 2) = 7

(- 3)^{2} - 4 \cdot 5 (- 2)

Apply rule

- (- a) = a

= (- 3)^{2} + 4 \cdot 5 \cdot 2

Apply exponent rule:

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

n

is even

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

= 3^{2} + 4 \cdot 5 \cdot 2

Multiply the numbers:

4 \cdot 5 \cdot 2 = 40

= 3^{2} + 40

3^{2} = 9

= 9 + 40

Add the numbers:

9 + 40 = 49

= 49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

7^{2} = 7

= 7

u_{1, 2} = \frac{- ( - 3 ) \pm 7}{2 \cdot 5}

Separate the solutions

u_{1} = \frac{- ( - 3 ) + 7}{2 \cdot 5}, u_{2} = \frac{- ( - 3 ) - 7}{2 \cdot 5}

u = \frac{- ( - 3 ) + 7}{2 \cdot 5} : 1

\frac{- ( - 3 ) + 7}{2 \cdot 5}

Apply rule

- (- a) = a

= \frac{3 + 7}{2 \cdot 5}

Add the numbers:

3 + 7 = 10

= \frac{10}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{10}{10}

Apply rule

\frac{a}{a} = 1

= 1

u = \frac{- ( - 3 ) - 7}{2 \cdot 5} : - \frac{2}{5}

\frac{- ( - 3 ) - 7}{2 \cdot 5}

Apply rule

- (- a) = a

= \frac{3 - 7}{2 \cdot 5}

Subtract the numbers:

3 - 7 = - 4

= \frac{- 4}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{- 4}{10}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{4}{10}

Cancel the common factor:

2

= - \frac{2}{5}

The solutions to the quadratic equation are:

u = 1, u = - \frac{2}{5}

Substitute back

u = sin (x)

sin (x) = 1, sin (x) = - \frac{2}{5}

sin (x) = 1, sin (x) = - \frac{2}{5}

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = 1

General solutions for

sin (x) = 1

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

sin (x) = - \frac{2}{5} : x = arcsin (- \frac{2}{5}) + 2 πn, x = π + arcsin (\frac{2}{5}) + 2 πn

sin (x) = - \frac{2}{5}

Apply trig inverse properties

sin (x) = - \frac{2}{5}

General solutions for

sin (x) = - \frac{2}{5}

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

x = arcsin (- \frac{2}{5}) + 2 πn, x = π + arcsin (\frac{2}{5}) + 2 πn

x = arcsin (- \frac{2}{5}) + 2 πn, x = π + arcsin (\frac{2}{5}) + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = arcsin (- \frac{2}{5}) + 2 πn, x = π + arcsin (\frac{2}{5}) + 2 πn

Show solutions in decimal form

x = \frac{π}{2} + 2 πn, x = - 0.41151 \dots + 2 πn, x = π + 0.41151 \dots + 2 πn

Popular Examples

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

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

solvefor α,sin^2(3α)=0.56 solve for

α, sin^{2} (3 α) = 0.56

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

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

tan(pi/2-x)+2tan(pi/2+x)=1

tan (\frac{π}{2} - x) + 2 tan (\frac{π}{2} + x) = 1

(11.0994)/(cos(x))=(3.5355)/(sin(x))

\frac{11.0994}{cos ( x )} = \frac{3.5355}{sin ( x )}

Frequently Asked Questions (FAQ)

What is the general solution for 5sin^2(x)=3sin(x)+2 ?
The general solution for 5sin^2(x)=3sin(x)+2 is x= pi/2+2pin,x=-0.41151…+2pin,x=pi+0.41151…+2pin