What is the general solution for 7sin^2(x)-14sin(x)+2=-5 ?

The general solution for 7sin^2(x)-14sin(x)+2=-5 is x= pi/2+2pin

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7sin^2(x)-14sin(x)+2=-5

Solution

7 sin^{2} (x) - 14 sin (x) + 2 = - 5

Solution

x = \frac{π}{2} + 2 πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n

Solution steps

7 sin^{2} (x) - 14 sin (x) + 2 = - 5

Solve by substitution

7 sin^{2} (x) - 14 sin (x) + 2 = - 5

Let:

sin (x) = u

7 u^{2} - 14 u + 2 = - 5

7 u^{2} - 14 u + 2 = - 5 : u = 1

7 u^{2} - 14 u + 2 = - 5

Move

5

to the left side

7 u^{2} - 14 u + 2 = - 5

Add

5

to both sides

7 u^{2} - 14 u + 2 + 5 = - 5 + 5

Simplify

7 u^{2} - 14 u + 7 = 0

7 u^{2} - 14 u + 7 = 0

Solve with the quadratic formula

7 u^{2} - 14 u + 7 = 0

Quadratic Equation Formula:

For

a = 7, b = - 14, c = 7

u_{1, 2} = \frac{- ( - 14 ) \pm ( - 14 ) ^{2} - 4 \cdot 7 \cdot 7}{2 \cdot 7}

u_{1, 2} = \frac{- ( - 14 ) \pm ( - 14 ) ^{2} - 4 \cdot 7 \cdot 7}{2 \cdot 7}

(- 14)^{2} - 4 \cdot 7 \cdot 7 = 0

(- 14)^{2} - 4 \cdot 7 \cdot 7

Apply exponent rule:

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

n

is even

(- 14)^{2} = 1 4^{2}

= 1 4^{2} - 4 \cdot 7 \cdot 7

Multiply the numbers:

4 \cdot 7 \cdot 7 = 196

= 1 4^{2} - 196

1 4^{2} = 196

= 196 - 196

Subtract the numbers:

196 - 196 = 0

= 0

u_{1, 2} = \frac{- ( - 14 ) \pm 0}{2 \cdot 7}

u = \frac{- ( - 14 )}{2 \cdot 7}

\frac{- ( - 14 )}{2 \cdot 7} = 1

\frac{- ( - 14 )}{2 \cdot 7}

Apply rule

- (- a) = a

= \frac{14}{2 \cdot 7}

Multiply the numbers:

2 \cdot 7 = 14

= \frac{14}{14}

Apply rule

\frac{a}{a} = 1

= 1

u = 1

The solution to the quadratic equation is:

u = 1

Substitute back

u = sin (x)

sin (x) = 1

sin (x) = 1

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

Combine all the solutions

x = \frac{π}{2} + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 7sin^2(x)-14sin(x)+2=-5 ?
The general solution for 7sin^2(x)-14sin(x)+2=-5 is x= pi/2+2pin