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

The general solution for 4sin(x/2)+2cos(x)=3 is x= pi/3+4pin,x=(5pi)/3+4pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

4sin(x/2)+2cos(x)=3

Solution

4 sin (\frac{x}{2}) + 2 cos (x) = 3

Solution

x = \frac{π}{3} + 4 πn, x = \frac{5 π}{3} + 4 πn

Degrees

x = 6 0^{\circ} + 72 0^{\circ} n, x = 30 0^{\circ} + 72 0^{\circ} n

Solution steps

4 sin (\frac{x}{2}) + 2 cos (x) = 3

Subtract

3

from both sides

4 sin (\frac{x}{2}) + 2 cos (x) - 3 = 0

Let:

u = \frac{x}{2}

4 sin (u) + 2 cos (2 u) - 3 = 0

Rewrite using trig identities

- 3 + 2 cos (2 u) + 4 sin (u)

Use the Double Angle identity:

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

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

Simplify

- 3 + 2 (1 - 2 sin^{2} (u)) + 4 sin (u) : 4 sin (u) - 4 sin^{2} (u) - 1

- 3 + 2 (1 - 2 sin^{2} (u)) + 4 sin (u)

Expand

2 (1 - 2 sin^{2} (u)) : 2 - 4 sin^{2} (u)

2 (1 - 2 sin^{2} (u))

Apply the distributive law:

a (b - c) = ab - a c

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

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

Simplify

2 \cdot 1 - 2 \cdot 2 sin^{2} (u) : 2 - 4 sin^{2} (u)

2 \cdot 1 - 2 \cdot 2 sin^{2} (u)

Multiply the numbers:

2 \cdot 1 = 2

= 2 - 2 \cdot 2 sin^{2} (u)

Multiply the numbers:

2 \cdot 2 = 4

= 2 - 4 sin^{2} (u)

= 2 - 4 sin^{2} (u)

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

Add/Subtract the numbers:

- 3 + 2 = - 1

= 4 sin (u) - 4 sin^{2} (u) - 1

= 4 sin (u) - 4 sin^{2} (u) - 1

- 1 + 4 sin (u) - 4 sin^{2} (u) = 0

Solve by substitution

- 1 + 4 sin (u) - 4 sin^{2} (u) = 0

Let:

sin (u) = u

- 1 + 4 u - 4 u^{2} = 0

- 1 + 4 u - 4 u^{2} = 0 : u = \frac{1}{2}

- 1 + 4 u - 4 u^{2} = 0

Write in the standard form

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

- 4 u^{2} + 4 u - 1 = 0

Solve with the quadratic formula

- 4 u^{2} + 4 u - 1 = 0

Quadratic Equation Formula:

For

a = - 4, b = 4, c = - 1

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

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

4^{2} - 4 (- 4) (- 1) = 0

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

Apply rule

- (- a) = a

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

Multiply the numbers:

4 \cdot 4 \cdot 1 = 16

= 4^{2} - 16

4^{2} = 16

= 16 - 16

Subtract the numbers:

16 - 16 = 0

= 0

u_{1, 2} = \frac{- 4 \pm 0}{2 ( - 4 )}

u = \frac{- 4}{2 ( - 4 )}

\frac{- 4}{2 ( - 4 )} = \frac{1}{2}

\frac{- 4}{2 ( - 4 )}

Remove parentheses:

(- a) = - a

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

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 4}{- 8}

Apply the fraction rule:

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

= \frac{4}{8}

Cancel the common factor:

4

= \frac{1}{2}

u = \frac{1}{2}

The solution to the quadratic equation is:

u = \frac{1}{2}

Substitute back

u = sin (u)

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

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

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

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

General solutions for

sin (u) = \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}

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

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

Combine all the solutions

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

Substitute back

u = \frac{x}{2}

\frac{x}{2} = \frac{π}{6} + 2 πn : x = \frac{π}{3} + 4 πn

\frac{x}{2} = \frac{π}{6} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{π}{6} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot \frac{π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{π}{6} + 2 \cdot 2 πn : \frac{π}{3} + 4 πn

2 \cdot \frac{π}{6} + 2 \cdot 2 πn

2 \cdot \frac{π}{6} = \frac{π}{3}

2 \cdot \frac{π}{6}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{π 2}{6}

Cancel the common factor:

2

= \frac{π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{π}{3} + 4 πn

x = \frac{π}{3} + 4 πn

x = \frac{π}{3} + 4 πn

x = \frac{π}{3} + 4 πn

\frac{x}{2} = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{3} + 4 πn

\frac{x}{2} = \frac{5 π}{6} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{5 π}{6} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn : \frac{5 π}{3} + 4 πn

2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{5 π}{6} = \frac{5 π}{3}

2 \cdot \frac{5 π}{6}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{5 π 2}{6}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{10 π}{6}

Cancel the common factor:

2

= \frac{5 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{5 π}{3} + 4 πn

x = \frac{5 π}{3} + 4 πn

x = \frac{5 π}{3} + 4 πn

x = \frac{5 π}{3} + 4 πn

x = \frac{π}{3} + 4 πn, x = \frac{5 π}{3} + 4 πn

Popular Examples

cos(x)= 1/5

cos (x) = \frac{1}{5}

sin(t)=(sqrt(3))/2

sin (t) = \frac{3}{2}

sin(θ)= 15/17

sin (θ) = \frac{15}{17}

sin(θ)=cos(2θ)

sin (θ) = cos (2 θ)

13sin(x)+6=sin(x)

13 sin (x) + 6 = sin (x)

Frequently Asked Questions (FAQ)

What is the general solution for 4sin(x/2)+2cos(x)=3 ?
The general solution for 4sin(x/2)+2cos(x)=3 is x= pi/3+4pin,x=(5pi)/3+4pin