What is the general solution for sin(x)+sin^2(x/2)= 1/2 ?

The general solution for sin(x)+sin^2(x/2)= 1/2 is x=0.46364…+pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

sin(x)+sin^2(x/2)= 1/2

Solution

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

Solution

x = 0.46364 \dots + πn

Degrees

x = 26.56505 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

\frac{1}{2}

from both sides

sin (x) + sin^{2} (\frac{x}{2}) - \frac{1}{2} = 0

Simplify

sin (x) + sin^{2} (\frac{x}{2}) - \frac{1}{2} : \frac{2 sin ( x ) + 2 sin ^{2} ( \frac{x}{2} ) - 1}{2}

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

Convert element to fraction:

sin (x) = \frac{sin ( x ) 2}{2}, sin^{2} (\frac{x}{2}) = \frac{sin ^{2} ( \frac{x}{2} ) 2}{2}

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

Since the denominators are equal, combine the fractions:

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

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

\frac{2 sin ( x ) + 2 sin ^{2} ( \frac{x}{2} ) - 1}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

2 sin (x) + 2 sin^{2} (\frac{x}{2}) - 1 = 0

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

= 2 sin (x) - cos (2 \cdot \frac{x}{2})

Multiply

2 \cdot \frac{x}{2} : x

2 \cdot \frac{x}{2}

Multiply fractions:

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

= \frac{x \cdot 2}{2}

Cancel the common factor:

2

= x

= 2 sin (x) - cos (x)

Divide both sides by

cos (x), cos (x) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

- 1 + 2 tan (x) = 0

- 1 + 2 tan (x) = 0

Move

1

to the right side

- 1 + 2 tan (x) = 0

Add

1

to both sides

- 1 + 2 tan (x) + 1 = 0 + 1

Simplify

2 tan (x) = 1

2 tan (x) = 1

Divide both sides by

2

2 tan (x) = 1

Divide both sides by

2

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

Simplify

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

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

Apply trig inverse properties

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

General solutions for

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

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (\frac{1}{2}) + πn

x = arctan (\frac{1}{2}) + πn

Show solutions in decimal form

x = 0.46364 \dots + πn

Popular Examples

sin^5(x)+sin^3(x)=0

sin^{5} (x) + sin^{3} (x) = 0

5sin^2(x)cos(7x)-cos(7x)=0

5 sin^{2} (x) cos (7 x) - cos (7 x) = 0

solvefor x,cos^2(x)*z^2-2cos^2(x)*z+1=0 solve for

x, cos^{2} (x) \cdot z^{2} - 2 cos^{2} (x) \cdot z + 1 = 0

sin(a)+sin(120+a)+sin(120-a)=0

sin (a) + sin (12 0^{\circ} + a) + sin (12 0^{\circ} - a) = 0

3sin(x)sin(x)=5cos(x)-2

3 sin (x) sin (x) = 5 cos (x) - 2

Frequently Asked Questions (FAQ)

What is the general solution for sin(x)+sin^2(x/2)= 1/2 ?
The general solution for sin(x)+sin^2(x/2)= 1/2 is x=0.46364…+pin