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Popular Trigonometry >

sin^2(2x)+cos^2(3x)=1

Solution

sin^{2} (2 x) + cos^{2} (3 x) = 1

Solution

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

Degrees

x = 0^{\circ} + 14 4^{\circ} n, x = 7 2^{\circ} + 14 4^{\circ} n, x = 3 6^{\circ} + 14 4^{\circ} n, x = 10 8^{\circ} + 14 4^{\circ} n

Solution steps

sin^{2} (2 x) + cos^{2} (3 x) = 1

Subtract

1

from both sides

sin^{2} (2 x) + cos^{2} (3 x) - 1 = 0

Rewrite using trig identities

- 1 + cos^{2} (3 x) + sin^{2} (2 x)

Use the Pythagorean identity:

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

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

= sin^{2} (2 x) - sin^{2} (3 x)

sin^{2} (2 x) - sin^{2} (3 x) = 0

Factor

sin^{2} (2 x) - sin^{2} (3 x) : (sin (2 x) + sin (3 x)) (sin (2 x) - sin (3 x))

sin^{2} (2 x) - sin^{2} (3 x)

Apply Difference of Two Squares Formula:

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

sin^{2} (2 x) - sin^{2} (3 x) = (sin (2 x) + sin (3 x)) (sin (2 x) - sin (3 x))

= (sin (2 x) + sin (3 x)) (sin (2 x) - sin (3 x))

(sin (2 x) + sin (3 x)) (sin (2 x) - sin (3 x)) = 0

Solving each part separately

sin (2 x) + sin (3 x) = 0 or sin (2 x) - sin (3 x) = 0

sin (2 x) + sin (3 x) = 0 : x = π + 4 πn, x = 3 π + 4 πn, x = \frac{4 πn}{5}, x = \frac{2 π}{5} + \frac{4 πn}{5}

sin (2 x) + sin (3 x) = 0

Rewrite using trig identities

sin (2 x) + sin (3 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 + 3 x}{2}) cos (\frac{2 x - 3 x}{2})

Simplify

2 sin (\frac{2 x + 3 x}{2}) cos (\frac{2 x - 3 x}{2}) : 2 cos (\frac{x}{2}) sin (\frac{5 x}{2})

2 sin (\frac{2 x + 3 x}{2}) cos (\frac{2 x - 3 x}{2})

Add similar elements:

2 x + 3 x = 5 x

= 2 sin (\frac{5 x}{2}) cos (\frac{2 x - 3 x}{2})

\frac{2 x - 3 x}{2} = - \frac{x}{2}

\frac{2 x - 3 x}{2}

Add similar elements:

2 x - 3 x = - x

= \frac{- x}{2}

Apply the fraction rule:

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

= - \frac{x}{2}

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

Use the negative angle identity:

cos (- x) = cos (x)

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

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

2 cos (\frac{x}{2}) sin (\frac{5 x}{2}) = 0

Solving each part separately

cos (\frac{x}{2}) = 0 or sin (\frac{5 x}{2}) = 0

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

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

General solutions for

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

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

Solve

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

\frac{x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

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

Simplify

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

x = π + 4 πn

x = π + 4 πn

x = π + 4 πn

Solve

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

\frac{x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

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

Simplify

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

x = 3 π + 4 πn

x = 3 π + 4 πn

x = 3 π + 4 πn

x = π + 4 πn, x = 3 π + 4 πn

sin (\frac{5 x}{2}) = 0 : x = \frac{4 πn}{5}, x = \frac{2 π}{5} + \frac{4 πn}{5}

sin (\frac{5 x}{2}) = 0

General solutions for

sin (\frac{5 x}{2}) = 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}

\frac{5 x}{2} = 0 + 2 πn, \frac{5 x}{2} = π + 2 πn

\frac{5 x}{2} = 0 + 2 πn, \frac{5 x}{2} = π + 2 πn

Solve

\frac{5 x}{2} = 0 + 2 πn : x = \frac{4 πn}{5}

\frac{5 x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{5 x}{2} = 2 πn

Multiply both sides by

2

\frac{5 x}{2} = 2 πn

Multiply both sides by

2

\frac{2 \cdot 5 x}{2} = 2 \cdot 2 πn

Simplify

5 x = 4 πn

5 x = 4 πn

Divide both sides by

5

5 x = 4 πn

Divide both sides by

5

\frac{5 x}{5} = \frac{4 πn}{5}

Simplify

x = \frac{4 πn}{5}

x = \frac{4 πn}{5}

Solve

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

\frac{5 x}{2} = π + 2 πn

Multiply both sides by

2

\frac{5 x}{2} = π + 2 πn

Multiply both sides by

2

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

Simplify

5 x = 2 π + 4 πn

5 x = 2 π + 4 πn

Divide both sides by

5

5 x = 2 π + 4 πn

Divide both sides by

5

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

Simplify

x = \frac{2 π}{5} + \frac{4 πn}{5}

x = \frac{2 π}{5} + \frac{4 πn}{5}

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

Combine all the solutions

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

sin (2 x) - sin (3 x) = 0 : x = \frac{π}{5} + \frac{4 πn}{5}, x = \frac{3 π}{5} + \frac{4 πn}{5}, x = 4 πn, x = 2 π + 4 πn

sin (2 x) - sin (3 x) = 0

Rewrite using trig identities

sin (2 x) - sin (3 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 - 3 x}{2}) cos (\frac{2 x + 3 x}{2})

Simplify

2 sin (\frac{2 x - 3 x}{2}) cos (\frac{2 x + 3 x}{2}) : - 2 cos (\frac{5 x}{2}) sin (\frac{x}{2})

2 sin (\frac{2 x - 3 x}{2}) cos (\frac{2 x + 3 x}{2})

\frac{2 x - 3 x}{2} = - \frac{x}{2}

\frac{2 x - 3 x}{2}

Add similar elements:

2 x - 3 x = - x

= \frac{- x}{2}

Apply the fraction rule:

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

= - \frac{x}{2}

= 2 sin (- \frac{x}{2}) cos (\frac{2 x + 3 x}{2})

Use the negative angle identity:

sin (- x) = - sin (x)

= 2 cos (\frac{2 x + 3 x}{2}) (- sin (\frac{x}{2}))

Remove parentheses:

(- a) = - a

= - 2 cos (\frac{2 x + 3 x}{2}) sin (\frac{x}{2})

Add similar elements:

2 x + 3 x = 5 x

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

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

- 2 cos (\frac{5 x}{2}) sin (\frac{x}{2}) = 0

Solving each part separately

cos (\frac{5 x}{2}) = 0 or sin (\frac{x}{2}) = 0

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

cos (\frac{5 x}{2}) = 0

General solutions for

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

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

Solve

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

\frac{5 x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{5 x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 \cdot 5 x}{2} : 5 x

\frac{2 \cdot 5 x}{2}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{10 x}{2}

Divide the numbers:

\frac{10}{2} = 5

= 5 x

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

5 x = π + 4 πn

5 x = π + 4 πn

5 x = π + 4 πn

Divide both sides by

5

5 x = π + 4 πn

Divide both sides by

5

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

Simplify

x = \frac{π}{5} + \frac{4 πn}{5}

x = \frac{π}{5} + \frac{4 πn}{5}

Solve

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

\frac{5 x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{5 x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 \cdot 5 x}{2} : 5 x

\frac{2 \cdot 5 x}{2}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{10 x}{2}

Divide the numbers:

\frac{10}{2} = 5

= 5 x

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

5 x = 3 π + 4 πn

5 x = 3 π + 4 πn

5 x = 3 π + 4 πn

Divide both sides by

5

5 x = 3 π + 4 πn

Divide both sides by

5

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

Simplify

x = \frac{3 π}{5} + \frac{4 πn}{5}

x = \frac{3 π}{5} + \frac{4 πn}{5}

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

sin (\frac{x}{2}) = 0 : x = 4 πn, x = 2 π + 4 πn

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

General solutions for

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

\frac{x}{2} = 0 + 2 πn, \frac{x}{2} = π + 2 πn

\frac{x}{2} = 0 + 2 πn, \frac{x}{2} = π + 2 πn

Solve

\frac{x}{2} = 0 + 2 πn : x = 4 πn

\frac{x}{2} = 0 + 2 πn

0 + 2 πn = 2 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

Solve

\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

x = 4 πn, x = 2 π + 4 πn

Combine all the solutions

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

Combine all the solutions

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

Merge Overlapping Intervals

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

Graph

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