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Popular Trigonometry >

sin(2x)=sin(0.5x)

Solution

sin (2 x) = sin (0.5 x)

Solution

x = \frac{2 π}{5} + \frac{8 πn}{5}, x = \frac{6 π}{5} + \frac{8 πn}{5}, x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

Degrees

x = 7 2^{\circ} + 28 8^{\circ} n, x = 21 6^{\circ} + 28 8^{\circ} n, x = 0^{\circ} + 48 0^{\circ} n, x = 24 0^{\circ} + 48 0^{\circ} n

Solution steps

sin (2 x) = sin (0.5 x)

Subtract

sin (0.5 x)

from both sides

sin (2 x) - sin (0.5 x) = 0

Rewrite using trig identities

- sin (0.5 x) + sin (2 x)

Use the Sum to Product identity:

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

= 2 sin (\frac{2 x - 0.5 x}{2}) cos (\frac{2 x + 0.5 x}{2})

Simplify

2 sin (\frac{2 x - 0.5 x}{2}) cos (\frac{2 x + 0.5 x}{2}) : 2 sin (0.75 x) cos (1.25 x)

2 sin (\frac{2 x - 0.5 x}{2}) cos (\frac{2 x + 0.5 x}{2})

Add similar elements:

2 x - 0.5 x = 1.5 x

= 2 sin (\frac{1.5 x}{2}) cos (\frac{2 x + 0.5 x}{2})

Add similar elements:

2 x + 0.5 x = 2.5 x

= 2 sin (\frac{1.5 x}{2}) cos (\frac{2.5 x}{2})

Divide the numbers:

\frac{1.5}{2} = 0.75

= 2 sin (0.75 x) cos (\frac{2.5 x}{2})

Divide the numbers:

\frac{2.5}{2} = 1.25

= 2 sin (0.75 x) cos (1.25 x)

= 2 sin (0.75 x) cos (1.25 x)

2 cos (1.25 x) sin (0.75 x) = 0

Solving each part separately

cos (1.25 x) = 0 or sin (0.75 x) = 0

cos (1.25 x) = 0 : x = \frac{2 π}{5} + \frac{8 πn}{5}, x = \frac{6 π}{5} + \frac{8 πn}{5}

cos (1.25 x) = 0

General solutions for

cos (1.25 x) = 0

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}

1.25 x = \frac{π}{2} + 2 πn, 1.25 x = \frac{3 π}{2} + 2 πn

1.25 x = \frac{π}{2} + 2 πn, 1.25 x = \frac{3 π}{2} + 2 πn

Solve

1.25 x = \frac{π}{2} + 2 πn : x = \frac{2 π}{5} + \frac{8 πn}{5}

1.25 x = \frac{π}{2} + 2 πn

Multiply both sides by

100

1.25 x = \frac{π}{2} + 2 πn

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

1.25 x \cdot 100 = \frac{π}{2} \cdot 100 + 2 πn \cdot 100

Refine

125 x = 50 π + 200 πn

125 x = 50 π + 200 πn

Divide both sides by

125

125 x = 50 π + 200 πn

Divide both sides by

125

\frac{125 x}{125} = \frac{50 π}{125} + \frac{200 πn}{125}

Simplify

\frac{125 x}{125} = \frac{50 π}{125} + \frac{200 πn}{125}

Simplify

\frac{125 x}{125} : x

\frac{125 x}{125}

Divide the numbers:

\frac{125}{125} = 1

= x

Simplify

\frac{50 π}{125} + \frac{200 πn}{125} : \frac{2 π}{5} + \frac{8 πn}{5}

\frac{50 π}{125} + \frac{200 πn}{125}

Cancel

\frac{50 π}{125} : \frac{2 π}{5}

\frac{50 π}{125}

Cancel the common factor:

25

= \frac{2 π}{5}

= \frac{2 π}{5} + \frac{200 πn}{125}

Cancel

\frac{200 πn}{125} : \frac{8 πn}{5}

\frac{200 πn}{125}

Cancel the common factor:

25

= \frac{8 πn}{5}

= \frac{2 π}{5} + \frac{8 πn}{5}

x = \frac{2 π}{5} + \frac{8 πn}{5}

x = \frac{2 π}{5} + \frac{8 πn}{5}

x = \frac{2 π}{5} + \frac{8 πn}{5}

Solve

1.25 x = \frac{3 π}{2} + 2 πn : x = \frac{6 π}{5} + \frac{8 πn}{5}

1.25 x = \frac{3 π}{2} + 2 πn

Multiply both sides by

100

1.25 x = \frac{3 π}{2} + 2 πn

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

1.25 x \cdot 100 = \frac{3 π}{2} \cdot 100 + 2 πn \cdot 100

Refine

125 x = 150 π + 200 πn

125 x = 150 π + 200 πn

Divide both sides by

125

125 x = 150 π + 200 πn

Divide both sides by

125

\frac{125 x}{125} = \frac{150 π}{125} + \frac{200 πn}{125}

Simplify

\frac{125 x}{125} = \frac{150 π}{125} + \frac{200 πn}{125}

Simplify

\frac{125 x}{125} : x

\frac{125 x}{125}

Divide the numbers:

\frac{125}{125} = 1

= x

Simplify

\frac{150 π}{125} + \frac{200 πn}{125} : \frac{6 π}{5} + \frac{8 πn}{5}

\frac{150 π}{125} + \frac{200 πn}{125}

Cancel

\frac{150 π}{125} : \frac{6 π}{5}

\frac{150 π}{125}

Cancel the common factor:

25

= \frac{6 π}{5}

= \frac{6 π}{5} + \frac{200 πn}{125}

Cancel

\frac{200 πn}{125} : \frac{8 πn}{5}

\frac{200 πn}{125}

Cancel the common factor:

25

= \frac{8 πn}{5}

= \frac{6 π}{5} + \frac{8 πn}{5}

x = \frac{6 π}{5} + \frac{8 πn}{5}

x = \frac{6 π}{5} + \frac{8 πn}{5}

x = \frac{6 π}{5} + \frac{8 πn}{5}

x = \frac{2 π}{5} + \frac{8 πn}{5}, x = \frac{6 π}{5} + \frac{8 πn}{5}

sin (0.75 x) = 0 : x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

sin (0.75 x) = 0

General solutions for

sin (0.75 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}

0.75 x = 0 + 2 πn, 0.75 x = π + 2 πn

0.75 x = 0 + 2 πn, 0.75 x = π + 2 πn

Solve

0.75 x = 0 + 2 πn : x = \frac{8 πn}{3}

0.75 x = 0 + 2 πn

0 + 2 πn = 2 πn

0.75 x = 2 πn

Multiply both sides by

100

0.75 x = 2 πn

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

0.75 x \cdot 100 = 2 πn \cdot 100

Refine

75 x = 200 πn

75 x = 200 πn

Divide both sides by

75

75 x = 200 πn

Divide both sides by

75

\frac{75 x}{75} = \frac{200 πn}{75}

Simplify

x = \frac{8 πn}{3}

x = \frac{8 πn}{3}

Solve

0.75 x = π + 2 πn : x = \frac{4 π}{3} + \frac{8 πn}{3}

0.75 x = π + 2 πn

Multiply both sides by

100

0.75 x = π + 2 πn

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

0.75 x \cdot 100 = π 100 + 2 πn \cdot 100

Refine

75 x = 100 π + 200 πn

75 x = 100 π + 200 πn

Divide both sides by

75

75 x = 100 π + 200 πn

Divide both sides by

75

\frac{75 x}{75} = \frac{100 π}{75} + \frac{200 πn}{75}

Simplify

\frac{75 x}{75} = \frac{100 π}{75} + \frac{200 πn}{75}

Simplify

\frac{75 x}{75} : x

\frac{75 x}{75}

Divide the numbers:

\frac{75}{75} = 1

= x

Simplify

\frac{100 π}{75} + \frac{200 πn}{75} : \frac{4 π}{3} + \frac{8 πn}{3}

\frac{100 π}{75} + \frac{200 πn}{75}

Cancel

\frac{100 π}{75} : \frac{4 π}{3}

\frac{100 π}{75}

Cancel the common factor:

25

= \frac{4 π}{3}

= \frac{4 π}{3} + \frac{200 πn}{75}

Cancel

\frac{200 πn}{75} : \frac{8 πn}{3}

\frac{200 πn}{75}

Cancel the common factor:

25

= \frac{8 πn}{3}

= \frac{4 π}{3} + \frac{8 πn}{3}

x = \frac{4 π}{3} + \frac{8 πn}{3}

x = \frac{4 π}{3} + \frac{8 πn}{3}

x = \frac{4 π}{3} + \frac{8 πn}{3}

x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

Combine all the solutions

x = \frac{2 π}{5} + \frac{8 πn}{5}, x = \frac{6 π}{5} + \frac{8 πn}{5}, x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

Graph

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