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Popular Trigonometry >

sin(5x-15)=sin(5*x)

Solution

sin (5 x - 15) = sin (5 \cdot x)

Solution

x = \frac{π}{10} + \frac{3}{2} + \frac{2 πn}{5}, x = \frac{3 π}{10} + \frac{3}{2} + \frac{2 πn}{5}

Degrees

x = 103.94366 \dots^{\circ} + 7 2^{\circ} n, x = 139.94366 \dots^{\circ} + 7 2^{\circ} n

Solution steps

sin (5 x - 15) = sin (5 x)

Subtract

sin (5 x)

from both sides

sin (5 x - 15) - sin (5 x) = 0

Rewrite using trig identities

sin (5 x - 15) - sin (5 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{5 x - 15 - 5 x}{2}) cos (\frac{5 x - 15 + 5 x}{2})

Simplify

2 sin (\frac{5 x - 15 - 5 x}{2}) cos (\frac{5 x - 15 + 5 x}{2}) : - 2 sin (\frac{15}{2}) cos (\frac{10 x - 15}{2})

2 sin (\frac{5 x - 15 - 5 x}{2}) cos (\frac{5 x - 15 + 5 x}{2})

\frac{5 x - 15 - 5 x}{2} = - \frac{15}{2}

\frac{5 x - 15 - 5 x}{2}

5 x - 15 - 5 x = - 15

5 x - 15 - 5 x

Group like terms

= 5 x - 5 x - 15

Add similar elements:

5 x - 5 x = 0

= - 15

= \frac{- 15}{2}

Apply the fraction rule:

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

= - \frac{15}{2}

= 2 sin (- \frac{15}{2}) cos (\frac{5 x + 5 x - 15}{2})

Simplify

sin (- \frac{15}{2}) : - sin (\frac{15}{2})

sin (- \frac{15}{2})

Use the following property:

sin (- x) = - sin (x)

sin (- \frac{15}{2}) = - sin (\frac{15}{2})

= - sin (\frac{15}{2})

= 2 (- sin (\frac{15}{2})) cos (\frac{5 x + 5 x - 15}{2})

5 x - 15 + 5 x = 10 x - 15

5 x - 15 + 5 x

Group like terms

= 5 x + 5 x - 15

Add similar elements:

5 x + 5 x = 10 x

= 10 x - 15

= 2 (- sin (\frac{15}{2})) cos (\frac{10 x - 15}{2})

Remove parentheses:

(- a) = - a

= - 2 sin (\frac{15}{2}) cos (\frac{10 x - 15}{2})

= - 2 sin (\frac{15}{2}) cos (\frac{10 x - 15}{2})

- 2 sin (\frac{15}{2}) cos (\frac{10 x - 15}{2}) = 0

Divide both sides by

- 2 sin (\frac{15}{2})

- 2 sin (\frac{15}{2}) cos (\frac{10 x - 15}{2}) = 0

Divide both sides by

- 2 sin (\frac{15}{2})

\frac{- 2 sin ( \frac{15}{2} ) cos ( \frac{10 x - 15}{2} )}{- 2 sin ( \frac{15}{2} )} = \frac{0}{- 2 sin ( \frac{15}{2} )}

Simplify

cos (\frac{10 x - 15}{2}) = 0

cos (\frac{10 x - 15}{2}) = 0

General solutions for

cos (\frac{10 x - 15}{2}) = 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}

\frac{10 x - 15}{2} = \frac{π}{2} + 2 πn, \frac{10 x - 15}{2} = \frac{3 π}{2} + 2 πn

\frac{10 x - 15}{2} = \frac{π}{2} + 2 πn, \frac{10 x - 15}{2} = \frac{3 π}{2} + 2 πn

Solve

\frac{10 x - 15}{2} = \frac{π}{2} + 2 πn : x = \frac{π}{10} + \frac{3}{2} + \frac{2 πn}{5}

\frac{10 x - 15}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{10 x - 15}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{2 ( 10 x - 15 )}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( 10 x - 15 )}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( 10 x - 15 )}{2} : 10 x - 15

\frac{2 ( 10 x - 15 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 10 x - 15

Simplify

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

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

10 x - 15 = π + 4 πn

10 x - 15 = π + 4 πn

10 x - 15 = π + 4 πn

Move

15

to the right side

10 x - 15 = π + 4 πn

Add

15

to both sides

10 x - 15 + 15 = π + 4 πn + 15

Simplify

10 x = π + 4 πn + 15

10 x = π + 4 πn + 15

Divide both sides by

10

10 x = π + 4 πn + 15

Divide both sides by

10

\frac{10 x}{10} = \frac{π}{10} + \frac{4 πn}{10} + \frac{15}{10}

Simplify

\frac{10 x}{10} = \frac{π}{10} + \frac{4 πn}{10} + \frac{15}{10}

Simplify

\frac{10 x}{10} : x

\frac{10 x}{10}

Divide the numbers:

\frac{10}{10} = 1

= x

Simplify

\frac{π}{10} + \frac{4 πn}{10} + \frac{15}{10} : \frac{π}{10} + \frac{3}{2} + \frac{2 πn}{5}

\frac{π}{10} + \frac{4 πn}{10} + \frac{15}{10}

Group like terms

= \frac{π}{10} + \frac{15}{10} + \frac{4 πn}{10}

Cancel

\frac{15}{10} : \frac{3}{2}

\frac{15}{10}

Cancel the common factor:

5

= \frac{3}{2}

= \frac{π}{10} + \frac{3}{2} + \frac{4 πn}{10}

Cancel

\frac{4 πn}{10} : \frac{2 πn}{5}

\frac{4 πn}{10}

Cancel the common factor:

2

= \frac{2 πn}{5}

= \frac{π}{10} + \frac{3}{2} + \frac{2 πn}{5}

x = \frac{π}{10} + \frac{3}{2} + \frac{2 πn}{5}

x = \frac{π}{10} + \frac{3}{2} + \frac{2 πn}{5}

x = \frac{π}{10} + \frac{3}{2} + \frac{2 πn}{5}

Solve

\frac{10 x - 15}{2} = \frac{3 π}{2} + 2 πn : x = \frac{3 π}{10} + \frac{3}{2} + \frac{2 πn}{5}

\frac{10 x - 15}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{10 x - 15}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{2 ( 10 x - 15 )}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( 10 x - 15 )}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( 10 x - 15 )}{2} : 10 x - 15

\frac{2 ( 10 x - 15 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 10 x - 15

Simplify

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

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 2}{2}

Cancel the common factor:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

10 x - 15 = 3 π + 4 πn

10 x - 15 = 3 π + 4 πn

10 x - 15 = 3 π + 4 πn

Move

15

to the right side

10 x - 15 = 3 π + 4 πn

Add

15

to both sides

10 x - 15 + 15 = 3 π + 4 πn + 15

Simplify

10 x = 3 π + 4 πn + 15

10 x = 3 π + 4 πn + 15

Divide both sides by

10

10 x = 3 π + 4 πn + 15

Divide both sides by

10

\frac{10 x}{10} = \frac{3 π}{10} + \frac{4 πn}{10} + \frac{15}{10}

Simplify

\frac{10 x}{10} = \frac{3 π}{10} + \frac{4 πn}{10} + \frac{15}{10}

Simplify

\frac{10 x}{10} : x

\frac{10 x}{10}

Divide the numbers:

\frac{10}{10} = 1

= x

Simplify

\frac{3 π}{10} + \frac{4 πn}{10} + \frac{15}{10} : \frac{3 π}{10} + \frac{3}{2} + \frac{2 πn}{5}

\frac{3 π}{10} + \frac{4 πn}{10} + \frac{15}{10}

Group like terms

= \frac{3 π}{10} + \frac{15}{10} + \frac{4 πn}{10}

Cancel

\frac{15}{10} : \frac{3}{2}

\frac{15}{10}

Cancel the common factor:

5

= \frac{3}{2}

= \frac{3 π}{10} + \frac{3}{2} + \frac{4 πn}{10}

Cancel

\frac{4 πn}{10} : \frac{2 πn}{5}

\frac{4 πn}{10}

Cancel the common factor:

2

= \frac{2 πn}{5}

= \frac{3 π}{10} + \frac{3}{2} + \frac{2 πn}{5}

x = \frac{3 π}{10} + \frac{3}{2} + \frac{2 πn}{5}

x = \frac{3 π}{10} + \frac{3}{2} + \frac{2 πn}{5}

x = \frac{3 π}{10} + \frac{3}{2} + \frac{2 πn}{5}

x = \frac{π}{10} + \frac{3}{2} + \frac{2 πn}{5}, x = \frac{3 π}{10} + \frac{3}{2} + \frac{2 πn}{5}

Graph

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