What is the general solution for cos(2θ)=11sin(θ)-5 ?

The general solution for cos(2θ)=11sin(θ)-5 is θ= pi/6+2pin,θ=(5pi)/6+2pin

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cos(2θ)=11sin(θ)-5

Solution

cos (2 θ) = 11 sin (θ) - 5

Solution

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

Degrees

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

Solution steps

cos (2 θ) = 11 sin (θ) - 5

Subtract

11 sin (θ) - 5

from both sides

cos (2 θ) - 11 sin (θ) + 5 = 0

Rewrite using trig identities

5 + cos (2 θ) - 11 sin (θ)

Use the Double Angle identity:

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

= 5 + 1 - 2 sin^{2} (θ) - 11 sin (θ)

Simplify

= - 2 sin^{2} (θ) - 11 sin (θ) + 6

6 - 11 sin (θ) - 2 sin^{2} (θ) = 0

Solve by substitution

6 - 11 sin (θ) - 2 sin^{2} (θ) = 0

Let:

sin (θ) = u

6 - 11 u - 2 u^{2} = 0

6 - 11 u - 2 u^{2} = 0 : u = - 6, u = \frac{1}{2}

6 - 11 u - 2 u^{2} = 0

Write in the standard form

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

- 2 u^{2} - 11 u + 6 = 0

Solve with the quadratic formula

- 2 u^{2} - 11 u + 6 = 0

Quadratic Equation Formula:

For

a = - 2, b = - 11, c = 6

u_{1, 2} = \frac{- ( - 11 ) \pm ( - 11 ) ^{2} - 4 ( - 2 ) \cdot 6}{2 ( - 2 )}

u_{1, 2} = \frac{- ( - 11 ) \pm ( - 11 ) ^{2} - 4 ( - 2 ) \cdot 6}{2 ( - 2 )}

(- 11)^{2} - 4 (- 2) \cdot 6 = 13

(- 11)^{2} - 4 (- 2) \cdot 6

Apply rule

- (- a) = a

= (- 11)^{2} + 4 \cdot 2 \cdot 6

Apply exponent rule:

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

n

is even

(- 11)^{2} = 1 1^{2}

= 1 1^{2} + 4 \cdot 2 \cdot 6

Multiply the numbers:

4 \cdot 2 \cdot 6 = 48

= 1 1^{2} + 48

1 1^{2} = 121

= 121 + 48

Add the numbers:

121 + 48 = 169

= 169

Factor the number:

169 = 1 3^{2}

= 1 3^{2}

Apply radical rule:

n a^{n} = a

1 3^{2} = 13

= 13

u_{1, 2} = \frac{- ( - 11 ) \pm 13}{2 ( - 2 )}

Separate the solutions

u_{1} = \frac{- ( - 11 ) + 13}{2 ( - 2 )}, u_{2} = \frac{- ( - 11 ) - 13}{2 ( - 2 )}

u = \frac{- ( - 11 ) + 13}{2 ( - 2 )} : - 6

\frac{- ( - 11 ) + 13}{2 ( - 2 )}

Remove parentheses:

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

= \frac{11 + 13}{- 2 \cdot 2}

Add the numbers:

11 + 13 = 24

= \frac{24}{- 2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{24}{- 4}

Apply the fraction rule:

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

= - \frac{24}{4}

Divide the numbers:

\frac{24}{4} = 6

= - 6

u = \frac{- ( - 11 ) - 13}{2 ( - 2 )} : \frac{1}{2}

\frac{- ( - 11 ) - 13}{2 ( - 2 )}

Remove parentheses:

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

= \frac{11 - 13}{- 2 \cdot 2}

Subtract the numbers:

11 - 13 = - 2

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 2}{- 4}

Apply the fraction rule:

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

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

The solutions to the quadratic equation are:

u = - 6, u = \frac{1}{2}

Substitute back

u = sin (θ)

sin (θ) = - 6, sin (θ) = \frac{1}{2}

sin (θ) = - 6, sin (θ) = \frac{1}{2}

sin (θ) = - 6 :

No Solution

sin (θ) = - 6

- 1 \leq sin (x) \leq 1

No Solution

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

sin (θ) = \frac{1}{2}

General solutions for

sin (θ) = \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}

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

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

Combine all the solutions

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

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cos(2θ)=11sin(θ)-5 ?
The general solution for cos(2θ)=11sin(θ)-5 is θ= pi/6+2pin,θ=(5pi)/6+2pin