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

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

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Popular >

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

Solution

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

Solution

x = 2 πn, x = π + 2 πn

Degrees

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

Solution steps

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

Subtract

2 tan^{2} (x)

from both sides

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

Factor

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

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

Rewrite

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

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

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

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

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

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

(2)^{2} tan^{2} (x) = (2 tan (x))^{2}

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

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

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

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

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

Solving each part separately

sin (x) + 2 tan (x) = 0 or sin (x) - 2 tan (x) = 0

sin (x) + 2 tan (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) + 2 tan (x) = 0

Express with sin, cos

sin (x) + 2 tan (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply

2 \frac{sin ( x )}{cos ( x )} : \frac{2 sin ( x )}{cos ( x )}

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

Multiply fractions:

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

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

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

Convert element to fraction:

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

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

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

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

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

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

Factor

cos (x) sin (x) + sin (x) 2 : sin (x) (cos (x) + 2)

cos (x) sin (x) + sin (x) 2

Factor out common term

sin (x)

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

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

Solving each part separately

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

cos (x) + 2 = 0 :

No Solution

cos (x) + 2 = 0

Move

2

to the right side

cos (x) + 2 = 0

Subtract

2

from both sides

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

Simplify

cos (x) = - 2

cos (x) = - 2

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

x = 2 πn, x = π + 2 πn

sin (x) - 2 tan (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) - 2 tan (x) = 0

Express with sin, cos

sin (x) - 2 tan (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply

2 \frac{sin ( x )}{cos ( x )} : \frac{2 sin ( x )}{cos ( x )}

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

Multiply fractions:

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

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

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

Convert element to fraction:

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

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

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

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

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

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

Factor

cos (x) sin (x) - sin (x) 2 : sin (x) (cos (x) - 2)

cos (x) sin (x) - sin (x) 2

Factor out common term

sin (x)

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

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

Solving each part separately

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

cos (x) - 2 = 0 :

No Solution

cos (x) - 2 = 0

Move

2

to the right side

cos (x) - 2 = 0

Add

2

to both sides

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

Simplify

cos (x) = 2

cos (x) = 2

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

x = 2 πn, x = π + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn

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Frequently Asked Questions (FAQ)

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