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

The general solution for sin^2(x)=(tan(x))/2 is x=2pin,x=pi+2pin,x= pi/4+pin

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sin^2(x)=(tan(x))/2

Solution

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

Solution

x = 2 πn, x = π + 2 πn, x = \frac{π}{4} + πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

\frac{tan ( x )}{2}

from both sides

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

Simplify

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

sin^{2} (x) - \frac{tan ( x )}{2}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

2 sin^{2} (x) - tan (x) = 0

Express with sin, cos

- tan (x) + 2 sin^{2} (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

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

Factor

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

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

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

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

= - sin (x) + 2 sin (x) sin (x)

Factor out common term

sin (x)

= sin (x) (- 1 + 2 sin (x) cos (x))

sin (x) (- 1 + 2 sin (x) cos (x)) = 0

Solving each part separately

sin (x) = 0 or - 1 + 2 sin (x) cos (x) = 0

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

- 1 + 2 sin (x) cos (x) = 0 : x = \frac{π}{4} + πn

- 1 + 2 sin (x) cos (x) = 0

Rewrite using trig identities

- 1 + 2 sin (x) cos (x)

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

= - 1 + sin (2 x)

- 1 + sin (2 x) = 0

Move

1

to the right side

- 1 + sin (2 x) = 0

Add

1

to both sides

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

Simplify

sin (2 x) = 1

sin (2 x) = 1

General solutions for

sin (2 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}

2 x = \frac{π}{2} + 2 πn

2 x = \frac{π}{2} + 2 πn

Solve

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

2 x = \frac{π}{2} + 2 πn

Divide both sides by

2

2 x = \frac{π}{2} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{π}{2}}{2} + \frac{2 πn}{2} : \frac{π}{4} + πn

\frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{2}}{2} = \frac{π}{4}

\frac{\frac{π}{2}}{2}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π}{4}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{π}{4} + πn

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Frequently Asked Questions (FAQ)

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