What is the general solution for 4-4cos(2x)-3sin(x)=0 ?

The general solution for 4-4cos(2x)-3sin(x)=0 is x=0.38439…+2pin,x=pi-0.38439…+2pin,x=2pin,x=pi+2pin

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Popular >

4-4cos(2x)-3sin(x)=0

Solution

4 - 4 cos (2 x) - 3 sin (x) = 0

Solution

x = 0.38439 \dots + 2 πn, x = π - 0.38439 \dots + 2 πn, x = 2 πn, x = π + 2 πn

Degrees

x = 22.02431 \dots^{\circ} + 36 0^{\circ} n, x = 157.97568 \dots^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

4 - 4 cos (2 x) - 3 sin (x) = 0

Rewrite using trig identities

4 - 3 sin (x) - 4 cos (2 x)

Use the Double Angle identity:

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

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

Simplify

4 - 3 sin (x) - 4 (1 - 2 sin^{2} (x)) : 8 sin^{2} (x) - 3 sin (x)

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

Expand

- 4 (1 - 2 sin^{2} (x)) : - 4 + 8 sin^{2} (x)

- 4 (1 - 2 sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 4, b = 1, c = 2 sin^{2} (x)

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

Apply minus-plus rules

- (- a) = a

= - 4 \cdot 1 + 4 \cdot 2 sin^{2} (x)

Simplify

- 4 \cdot 1 + 4 \cdot 2 sin^{2} (x) : - 4 + 8 sin^{2} (x)

- 4 \cdot 1 + 4 \cdot 2 sin^{2} (x)

Multiply the numbers:

4 \cdot 1 = 4

= - 4 + 4 \cdot 2 sin^{2} (x)

Multiply the numbers:

4 \cdot 2 = 8

= - 4 + 8 sin^{2} (x)

= - 4 + 8 sin^{2} (x)

= 4 - 3 sin (x) - 4 + 8 sin^{2} (x)

Simplify

4 - 3 sin (x) - 4 + 8 sin^{2} (x) : 8 sin^{2} (x) - 3 sin (x)

4 - 3 sin (x) - 4 + 8 sin^{2} (x)

Group like terms

= - 3 sin (x) + 8 sin^{2} (x) + 4 - 4

4 - 4 = 0

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

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

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

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

Solve by substitution

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

Let:

sin (x) = u

- 3 u + 8 u^{2} = 0

- 3 u + 8 u^{2} = 0 : u = \frac{3}{8}, u = 0

- 3 u + 8 u^{2} = 0

Write in the standard form

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

8 u^{2} - 3 u = 0

Solve with the quadratic formula

8 u^{2} - 3 u = 0

Quadratic Equation Formula:

For

a = 8, b = - 3, c = 0

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 8 \cdot 0}{2 \cdot 8}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 8 \cdot 0}{2 \cdot 8}

(- 3)^{2} - 4 \cdot 8 \cdot 0 = 3

(- 3)^{2} - 4 \cdot 8 \cdot 0

Apply exponent rule:

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

n

is even

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

= 3^{2} - 4 \cdot 8 \cdot 0

Apply rule

0 \cdot a = 0

= 3^{2} - 0

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

= 3^{2}

Apply radical rule:

assuming

a \geq 0

= 3

u_{1, 2} = \frac{- ( - 3 ) \pm 3}{2 \cdot 8}

Separate the solutions

u_{1} = \frac{- ( - 3 ) + 3}{2 \cdot 8}, u_{2} = \frac{- ( - 3 ) - 3}{2 \cdot 8}

u = \frac{- ( - 3 ) + 3}{2 \cdot 8} : \frac{3}{8}

\frac{- ( - 3 ) + 3}{2 \cdot 8}

Apply rule

- (- a) = a

= \frac{3 + 3}{2 \cdot 8}

Add the numbers:

3 + 3 = 6

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

Multiply the numbers:

2 \cdot 8 = 16

= \frac{6}{16}

Cancel the common factor:

2

= \frac{3}{8}

u = \frac{- ( - 3 ) - 3}{2 \cdot 8} : 0

\frac{- ( - 3 ) - 3}{2 \cdot 8}

Apply rule

- (- a) = a

= \frac{3 - 3}{2 \cdot 8}

Subtract the numbers:

3 - 3 = 0

= \frac{0}{2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{0}{16}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

The solutions to the quadratic equation are:

u = \frac{3}{8}, u = 0

Substitute back

u = sin (x)

sin (x) = \frac{3}{8}, sin (x) = 0

sin (x) = \frac{3}{8}, sin (x) = 0

sin (x) = \frac{3}{8} : x = arcsin (\frac{3}{8}) + 2 πn, x = π - arcsin (\frac{3}{8}) + 2 πn

sin (x) = \frac{3}{8}

Apply trig inverse properties

sin (x) = \frac{3}{8}

General solutions for

sin (x) = \frac{3}{8}

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

x = arcsin (\frac{3}{8}) + 2 πn, x = π - arcsin (\frac{3}{8}) + 2 πn

x = arcsin (\frac{3}{8}) + 2 πn, x = π - arcsin (\frac{3}{8}) + 2 πn

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

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 = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

Combine all the solutions

x = arcsin (\frac{3}{8}) + 2 πn, x = π - arcsin (\frac{3}{8}) + 2 πn, x = 2 πn, x = π + 2 πn

Show solutions in decimal form

x = 0.38439 \dots + 2 πn, x = π - 0.38439 \dots + 2 πn, x = 2 πn, x = π + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 4-4cos(2x)-3sin(x)=0 ?
The general solution for 4-4cos(2x)-3sin(x)=0 is x=0.38439…+2pin,x=pi-0.38439…+2pin,x=2pin,x=pi+2pin