What is the general solution for cos(2x)=2+3sin(x), pi/2 <= x<= (3pi)/2 ?

Q: What is the general solution for cos(2x)=2+3sin(x), pi/2 <= x<= (3pi)/2 ?

The general solution for cos(2x)=2+3sin(x), pi/2 <= x<= (3pi)/2 is x=(3pi)/2 ,x=(7pi)/6

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cos(2x)=2+3sin(x), pi/2 <= x<= (3pi)/2

Solution

cos (2 x) = 2 + 3 sin (x), \frac{π}{2} \leq x \leq \frac{3 π}{2}

Solution

x = \frac{3 π}{2}, x = \frac{7 π}{6}

Degrees

x = 27 0^{\circ}, x = 21 0^{\circ}

Solution steps

cos (2 x) = 2 + 3 sin (x), \frac{π}{2} \leq x \leq \frac{3 π}{2}

Subtract

2 + 3 sin (x)

from both sides

cos (2 x) - 2 - 3 sin (x) = 0

Rewrite using trig identities

- 2 + cos (2 x) - 3 sin (x)

Use the Double Angle identity:

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

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

Simplify

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

- 1 - 2 sin^{2} (x) - 3 sin (x) = 0

Solve by substitution

- 1 - 2 sin^{2} (x) - 3 sin (x) = 0

Let:

sin (x) = u

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

- 1 - 2 u^{2} - 3 u = 0 : u = - 1, u = - \frac{1}{2}

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = - 2, b = - 3, c = - 1

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

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

(- 3)^{2} - 4 (- 2) (- 1) = 1

(- 3)^{2} - 4 (- 2) (- 1)

Apply rule

- (- a) = a

= (- 3)^{2} - 4 \cdot 2 \cdot 1

Apply exponent rule:

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

n

is even

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

= 3^{2} - 4 \cdot 2 \cdot 1

Multiply the numbers:

4 \cdot 2 \cdot 1 = 8

= 3^{2} - 8

3^{2} = 9

= 9 - 8

Subtract the numbers:

9 - 8 = 1

= 1

Apply rule

1 = 1

= 1

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

Separate the solutions

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

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

\frac{- ( - 3 ) + 1}{2 ( - 2 )}

Remove parentheses:

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

= \frac{3 + 1}{- 2 \cdot 2}

Add the numbers:

3 + 1 = 4

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4}{- 4}

Apply the fraction rule:

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

= - \frac{4}{4}

Apply rule

\frac{a}{a} = 1

= - 1

u = \frac{- ( - 3 ) - 1}{2 ( - 2 )} : - \frac{1}{2}

\frac{- ( - 3 ) - 1}{2 ( - 2 )}

Remove parentheses:

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

= \frac{3 - 1}{- 2 \cdot 2}

Subtract the numbers:

3 - 1 = 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 = - 1, u = - \frac{1}{2}

Substitute back

u = sin (x)

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

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

sin (x) = - 1, \frac{π}{2} \leq x \leq \frac{3 π}{2} : x = \frac{3 π}{2}

sin (x) = - 1, \frac{π}{2} \leq x \leq \frac{3 π}{2}

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{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

Solutions for the range

\frac{π}{2} \leq x \leq \frac{3 π}{2}

x = \frac{3 π}{2}

sin (x) = - \frac{1}{2}, \frac{π}{2} \leq x \leq \frac{3 π}{2} : x = \frac{7 π}{6}

sin (x) = - \frac{1}{2}, \frac{π}{2} \leq x \leq \frac{3 π}{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{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Solutions for the range

\frac{π}{2} \leq x \leq \frac{3 π}{2}

x = \frac{7 π}{6}

Combine all the solutions

x = \frac{3 π}{2}, x = \frac{7 π}{6}

Popular Examples

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tan (θ) = (\frac{3}{8})

29= 1/2 (5.6)(7.1)sin(x)

29 = \frac{1}{2} (5.6) (7.1) sin (x)

(10.4)/(sin(x))=(12)/(sin(62))

\frac{10.4}{sin ( x )} = \frac{12}{sin ( 6 2 ^{\circ} )}

csc(θ)=-2,0<= θ<= 360

csc (θ) = - 2, 0^{\circ} \leq θ \leq 36 0^{\circ}

sin(B)= 12/13

sin (B) = \frac{12}{13}

Frequently Asked Questions (FAQ)

What is the general solution for cos(2x)=2+3sin(x), pi/2 <= x<= (3pi)/2 ?
The general solution for cos(2x)=2+3sin(x), pi/2 <= x<= (3pi)/2 is x=(3pi)/2 ,x=(7pi)/6