What is the general solution for 8(cos(x))^2-6sin(x)=3 ?

The general solution for 8(cos(x))^2-6sin(x)=3 is x= pi/6+2pin,x=(5pi)/6+2pin

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8(cos(x))^2-6sin(x)=3

Solution

8 (cos (x))^{2} - 6 sin (x) = 3

Solution

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

8 (cos (x))^{2} - 6 sin (x) = 3

Subtract

3

from both sides

8 cos^{2} (x) - 6 sin (x) - 3 = 0

Rewrite using trig identities

- 3 - 6 sin (x) + 8 cos^{2} (x)

Use the Pythagorean identity:

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

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

= - 3 - 6 sin (x) + 8 (1 - sin^{2} (x))

Simplify

- 3 - 6 sin (x) + 8 (1 - sin^{2} (x)) : - 8 sin^{2} (x) - 6 sin (x) + 5

- 3 - 6 sin (x) + 8 (1 - sin^{2} (x))

Expand

8 (1 - sin^{2} (x)) : 8 - 8 sin^{2} (x)

8 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 8, b = 1, c = sin^{2} (x)

= 8 \cdot 1 - 8 sin^{2} (x)

Multiply the numbers:

8 \cdot 1 = 8

= 8 - 8 sin^{2} (x)

= - 3 - 6 sin (x) + 8 - 8 sin^{2} (x)

Simplify

- 3 - 6 sin (x) + 8 - 8 sin^{2} (x) : - 8 sin^{2} (x) - 6 sin (x) + 5

- 3 - 6 sin (x) + 8 - 8 sin^{2} (x)

Group like terms

= - 6 sin (x) - 8 sin^{2} (x) - 3 + 8

Add/Subtract the numbers:

- 3 + 8 = 5

= - 8 sin^{2} (x) - 6 sin (x) + 5

= - 8 sin^{2} (x) - 6 sin (x) + 5

= - 8 sin^{2} (x) - 6 sin (x) + 5

5 - 6 sin (x) - 8 sin^{2} (x) = 0

Solve by substitution

5 - 6 sin (x) - 8 sin^{2} (x) = 0

Let:

sin (x) = u

5 - 6 u - 8 u^{2} = 0

5 - 6 u - 8 u^{2} = 0 : u = - \frac{5}{4}, u = \frac{1}{2}

5 - 6 u - 8 u^{2} = 0

Write in the standard form

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

- 8 u^{2} - 6 u + 5 = 0

Solve with the quadratic formula

- 8 u^{2} - 6 u + 5 = 0

Quadratic Equation Formula:

For

a = - 8, b = - 6, c = 5

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 ( - 8 ) \cdot 5}{2 ( - 8 )}

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 ( - 8 ) \cdot 5}{2 ( - 8 )}

(- 6)^{2} - 4 (- 8) \cdot 5 = 14

(- 6)^{2} - 4 (- 8) \cdot 5

Apply rule

- (- a) = a

= (- 6)^{2} + 4 \cdot 8 \cdot 5

Apply exponent rule:

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

n

is even

(- 6)^{2} = 6^{2}

= 6^{2} + 4 \cdot 8 \cdot 5

Multiply the numbers:

4 \cdot 8 \cdot 5 = 160

= 6^{2} + 160

6^{2} = 36

= 36 + 160

Add the numbers:

36 + 160 = 196

= 196

Factor the number:

196 = 1 4^{2}

= 1 4^{2}

Apply radical rule:

1 4^{2} = 14

= 14

u_{1, 2} = \frac{- ( - 6 ) \pm 14}{2 ( - 8 )}

Separate the solutions

u_{1} = \frac{- ( - 6 ) + 14}{2 ( - 8 )}, u_{2} = \frac{- ( - 6 ) - 14}{2 ( - 8 )}

u = \frac{- ( - 6 ) + 14}{2 ( - 8 )} : - \frac{5}{4}

\frac{- ( - 6 ) + 14}{2 ( - 8 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{6 + 14}{- 2 \cdot 8}

Add the numbers:

6 + 14 = 20

= \frac{20}{- 2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{20}{- 16}

Apply the fraction rule:

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

= - \frac{20}{16}

Cancel the common factor:

4

= - \frac{5}{4}

u = \frac{- ( - 6 ) - 14}{2 ( - 8 )} : \frac{1}{2}

\frac{- ( - 6 ) - 14}{2 ( - 8 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{6 - 14}{- 2 \cdot 8}

Subtract the numbers:

6 - 14 = - 8

= \frac{- 8}{- 2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{- 8}{- 16}

Apply the fraction rule:

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

= \frac{8}{16}

Cancel the common factor:

8

= \frac{1}{2}

The solutions to the quadratic equation are:

u = - \frac{5}{4}, u = \frac{1}{2}

Substitute back

u = sin (x)

sin (x) = - \frac{5}{4}, sin (x) = \frac{1}{2}

sin (x) = - \frac{5}{4}, sin (x) = \frac{1}{2}

sin (x) = - \frac{5}{4} :

No Solution

sin (x) = - \frac{5}{4}

- 1 \leq sin (x) \leq 1

No Solution

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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

General solutions for

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

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}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Popular Examples

sin(x)=(-tan(x))/2

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

2sin(3θ)-1=0,0<= θ<= 2pi

2 sin (3 θ) - 1 = 0, 0 \leq θ \leq 2 π

sin(θ)=-5/7

sin (θ) = - \frac{5}{7}

cos(x)sin(x)=sin(x)

cos (x) sin (x) = sin (x)

sec^2(x)-1=1

sec^{2} (x) - 1 = 1

Frequently Asked Questions (FAQ)

What is the general solution for 8(cos(x))^2-6sin(x)=3 ?
The general solution for 8(cos(x))^2-6sin(x)=3 is x= pi/6+2pin,x=(5pi)/6+2pin