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Popular >

sin(x)=tan(x/2)

Solution

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

Solution

x = 4 πn, x = 2 π + 4 πn, x = \frac{3 π}{2} + 4 πn, x = \frac{5 π}{2} + 4 πn, x = \frac{π}{2} + 4 πn, x = \frac{7 π}{2} + 4 πn

Degrees

x = 0^{\circ} + 72 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n, x = 27 0^{\circ} + 72 0^{\circ} n, x = 45 0^{\circ} + 72 0^{\circ} n, x = 9 0^{\circ} + 72 0^{\circ} n, x = 63 0^{\circ} + 72 0^{\circ} n

Solution steps

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

Subtract

tan (\frac{x}{2})

from both sides

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

Let:

u = \frac{x}{2}

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

Express with sin, cos

sin (2 u) - tan (u)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

= - sin (u) + cos (u) \cdot 2 sin (u) cos (u)

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

cos (u) \cdot 2 sin (u) cos (u)

Apply exponent rule:

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

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

= 2 sin (u) cos^{1 + 1} (u)

Add the numbers:

1 + 1 = 2

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

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

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

Factor

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

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

Factor out common term

sin (u)

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

Factor

2 cos^{2} (u) - 1 : (2 cos (u) + 1) (2 cos (u) - 1)

2 cos^{2} (u) - 1

Rewrite

2 cos^{2} (u) - 1

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

2 cos^{2} (u) - 1

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

= (2)^{2} cos^{2} (u) - 1

Rewrite

1

1^{2}

= (2)^{2} cos^{2} (u) - 1^{2}

Apply exponent rule:

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

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

= (2 cos (u))^{2} - 1^{2}

= (2 cos (u))^{2} - 1^{2}

Apply Difference of Two Squares Formula:

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

(2 cos (u))^{2} - 1^{2} = (2 cos (u) + 1) (2 cos (u) - 1)

= (2 cos (u) + 1) (2 cos (u) - 1)

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

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

Solving each part separately

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

sin (u) = 0 : u = 2 πn, u = π + 2 πn

sin (u) = 0

General solutions for

sin (u) = 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}

u = 0 + 2 πn, u = π + 2 πn

u = 0 + 2 πn, u = π + 2 πn

Solve

u = 0 + 2 πn : u = 2 πn

u = 0 + 2 πn

0 + 2 πn = 2 πn

u = 2 πn

u = 2 πn, u = π + 2 πn

2 cos (u) + 1 = 0 : u = \frac{3 π}{4} + 2 πn, u = \frac{5 π}{4} + 2 πn

2 cos (u) + 1 = 0

Move

1

to the right side

2 cos (u) + 1 = 0

Subtract

1

from both sides

2 cos (u) + 1 - 1 = 0 - 1

Simplify

2 cos (u) = - 1

2 cos (u) = - 1

Divide both sides by

2

2 cos (u) = - 1

Divide both sides by

2

\frac{2 cos ( u )}{2} = \frac{- 1}{2}

Simplify

\frac{2 cos ( u )}{2} = \frac{- 1}{2}

Simplify

\frac{2 cos ( u )}{2} : cos (u)

\frac{2 cos ( u )}{2}

Cancel the common factor:

2

= cos (u)

Simplify

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

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

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

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

General solutions for

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

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

u = \frac{3 π}{4} + 2 πn, u = \frac{5 π}{4} + 2 πn

u = \frac{3 π}{4} + 2 πn, u = \frac{5 π}{4} + 2 πn

2 cos (u) - 1 = 0 : u = \frac{π}{4} + 2 πn, u = \frac{7 π}{4} + 2 πn

2 cos (u) - 1 = 0

Move

1

to the right side

2 cos (u) - 1 = 0

Add

1

to both sides

2 cos (u) - 1 + 1 = 0 + 1

Simplify

2 cos (u) = 1

2 cos (u) = 1

Divide both sides by

2

2 cos (u) = 1

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 cos ( u )}{2} : cos (u)

\frac{2 cos ( u )}{2}

Cancel the common factor:

2

= cos (u)

Simplify

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

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

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

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

General solutions for

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

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

u = \frac{π}{4} + 2 πn, u = \frac{7 π}{4} + 2 πn

u = \frac{π}{4} + 2 πn, u = \frac{7 π}{4} + 2 πn

Combine all the solutions

u = 2 πn, u = π + 2 πn, u = \frac{3 π}{4} + 2 πn, u = \frac{5 π}{4} + 2 πn, u = \frac{π}{4} + 2 πn, u = \frac{7 π}{4} + 2 πn

Substitute back

u = \frac{x}{2}

\frac{x}{2} = 2 πn : x = 4 πn

\frac{x}{2} = 2 πn

Multiply both sides by

2

\frac{x}{2} = 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot 2 πn

Simplify

x = 4 πn

x = 4 πn

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

\frac{x}{2} = π + 2 πn

Multiply both sides by

2

\frac{x}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 π + 2 \cdot 2 πn

Simplify

x = 2 π + 4 πn

x = 2 π + 4 πn

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

\frac{x}{2} = \frac{3 π}{4} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{3 π}{4} + 2 πn

Multiply both sides by

2

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

Simplify

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

2 \cdot \frac{3 π}{4} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{4} = \frac{3 π}{2}

2 \cdot \frac{3 π}{4}

Multiply fractions:

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

= \frac{3 π 2}{4}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 π}{4}

Cancel the common factor:

2

= \frac{3 π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{3 π}{2} + 4 πn

x = \frac{3 π}{2} + 4 πn

x = \frac{3 π}{2} + 4 πn

x = \frac{3 π}{2} + 4 πn

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

\frac{x}{2} = \frac{5 π}{4} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{5 π}{4} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{5 π}{4} + 2 \cdot 2 πn

Simplify

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

2 \cdot \frac{5 π}{4} + 2 \cdot 2 πn

2 \cdot \frac{5 π}{4} = \frac{5 π}{2}

2 \cdot \frac{5 π}{4}

Multiply fractions:

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

= \frac{5 π 2}{4}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{10 π}{4}

Cancel the common factor:

2

= \frac{5 π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{5 π}{2} + 4 πn

x = \frac{5 π}{2} + 4 πn

x = \frac{5 π}{2} + 4 πn

x = \frac{5 π}{2} + 4 πn

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

\frac{x}{2} = \frac{π}{4} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{π}{4} + 2 πn

Multiply both sides by

2

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

Simplify

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

2 \cdot \frac{π}{4} + 2 \cdot 2 πn

2 \cdot \frac{π}{4} = \frac{π}{2}

2 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{π}{2} + 4 πn

x = \frac{π}{2} + 4 πn

x = \frac{π}{2} + 4 πn

x = \frac{π}{2} + 4 πn

\frac{x}{2} = \frac{7 π}{4} + 2 πn : x = \frac{7 π}{2} + 4 πn

\frac{x}{2} = \frac{7 π}{4} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{7 π}{4} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{7 π}{4} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot \frac{7 π}{4} + 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{7 π}{4} + 2 \cdot 2 πn : \frac{7 π}{2} + 4 πn

2 \cdot \frac{7 π}{4} + 2 \cdot 2 πn

2 \cdot \frac{7 π}{4} = \frac{7 π}{2}

2 \cdot \frac{7 π}{4}

Multiply fractions:

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

= \frac{7 π 2}{4}

Multiply the numbers:

7 \cdot 2 = 14

= \frac{14 π}{4}

Cancel the common factor:

2

= \frac{7 π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{7 π}{2} + 4 πn

x = \frac{7 π}{2} + 4 πn

x = \frac{7 π}{2} + 4 πn

x = \frac{7 π}{2} + 4 πn

x = 4 πn, x = 2 π + 4 πn, x = \frac{3 π}{2} + 4 πn, x = \frac{5 π}{2} + 4 πn, x = \frac{π}{2} + 4 πn, x = \frac{7 π}{2} + 4 πn

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