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

The general solution for tan(x)sin^2(x)=sin(2x) is x=2pin,x=pi+2pin,x=-0.95531…+pin,x=0.95531…+pin

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Popular Trigonometry >

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

Solution

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

Solution

x = 2 πn, x = π + 2 πn, x = - 0.95531 \dots + πn, x = 0.95531 \dots + πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = - 54.73561 \dots^{\circ} + 18 0^{\circ} n, x = 54.73561 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

sin (2 x)

from both sides

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

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

Factor

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

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

Apply exponent rule:

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

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

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

Factor out common term

sin (x)

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

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

Solving each part separately

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

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

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Multiply fractions:

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

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

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

sin (x) sin (x)

Apply exponent rule:

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

sin (x) sin (x) = sin^{1 + 1} (x)

= sin^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= sin^{2} (x)

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

2 cos (x) cos (x)

Apply exponent rule:

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

cos (x) cos (x) = cos^{1 + 1} (x)

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

Add the numbers:

1 + 1 = 2

= 2 cos^{2} (x)

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

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

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

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

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

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

Factor

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

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

Rewrite

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

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

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

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

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

Apply exponent rule:

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

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

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

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

Apply Difference of Two Squares Formula:

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

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

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

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

Solving each part separately

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

sin (x) + 2 cos (x) = 0 : x = arctan (- 2) + πn

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

tan (x) + 2 = 0

tan (x) + 2 = 0

Move

2

to the right side

tan (x) + 2 = 0

Subtract

2

from both sides

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

Simplify

tan (x) = - 2

tan (x) = - 2

Apply trig inverse properties

tan (x) = - 2

General solutions for

tan (x) = - 2

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

x = arctan (- 2) + πn

x = arctan (- 2) + πn

sin (x) - 2 cos (x) = 0 : x = arctan (2) + πn

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

tan (x) - 2 = 0

tan (x) - 2 = 0

Move

2

to the right side

tan (x) - 2 = 0

Add

2

to both sides

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

Simplify

tan (x) = 2

tan (x) = 2

Apply trig inverse properties

tan (x) = 2

General solutions for

tan (x) = 2

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

x = arctan (2) + πn

x = arctan (2) + πn

Combine all the solutions

x = arctan (- 2) + πn, x = arctan (2) + πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arctan (- 2) + πn, x = arctan (2) + πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = - 0.95531 \dots + πn, x = 0.95531 \dots + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for tan(x)sin^2(x)=sin(2x) ?
The general solution for tan(x)sin^2(x)=sin(2x) is x=2pin,x=pi+2pin,x=-0.95531…+pin,x=0.95531…+pin