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Popular Trigonometry >

1-cos^2(a)=sin^2(2a)

Solution

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

Solution

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

Degrees

a = 0^{\circ} + 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n, a = 12 0^{\circ} + 36 0^{\circ} n, a = 24 0^{\circ} + 36 0^{\circ} n, a = 6 0^{\circ} + 24 0^{\circ} n, a = 18 0^{\circ} + 24 0^{\circ} n

Solution steps

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

Subtract

sin^{2} (2 a)

from both sides

1 - cos^{2} (a) - sin^{2} (2 a) = 0

Rewrite using trig identities

1 - cos^{2} (a) - sin^{2} (2 a)

Use the Pythagorean identity:

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

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

= - sin^{2} (2 a) + sin^{2} (a)

- sin^{2} (2 a) + sin^{2} (a) = 0

Factor

- sin^{2} (2 a) + sin^{2} (a) : (sin (a) + sin (2 a)) (sin (a) - sin (2 a))

- sin^{2} (2 a) + sin^{2} (a)

Apply Difference of Two Squares Formula:

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

sin^{2} (a) - sin^{2} (2 a) = (sin (a) + sin (2 a)) (sin (a) - sin (2 a))

= (sin (a) + sin (2 a)) (sin (a) - sin (2 a))

(sin (a) + sin (2 a)) (sin (a) - sin (2 a)) = 0

Solving each part separately

sin (a) + sin (2 a) = 0 or sin (a) - sin (2 a) = 0

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

sin (a) + sin (2 a) = 0

Rewrite using trig identities

sin (2 a) + sin (a)

Use the Double Angle identity:

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

= 2 sin (a) cos (a) + sin (a)

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

Factor

sin (a) + 2 cos (a) sin (a) : sin (a) (2 cos (a) + 1)

sin (a) + 2 cos (a) sin (a)

Factor out common term

sin (a)

= sin (a) (1 + 2 cos (a))

sin (a) (2 cos (a) + 1) = 0

Solving each part separately

sin (a) = 0 or 2 cos (a) + 1 = 0

sin (a) = 0 : a = 2 πn, a = π + 2 πn

sin (a) = 0

General solutions for

sin (a) = 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}

a = 0 + 2 πn, a = π + 2 πn

a = 0 + 2 πn, a = π + 2 πn

Solve

a = 0 + 2 πn : a = 2 πn

a = 0 + 2 πn

0 + 2 πn = 2 πn

a = 2 πn

a = 2 πn, a = π + 2 πn

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

2 cos (a) + 1 = 0

Move

1

to the right side

2 cos (a) + 1 = 0

Subtract

1

from both sides

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

Simplify

2 cos (a) = - 1

2 cos (a) = - 1

Divide both sides by

2

2 cos (a) = - 1

Divide both sides by

2

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

Simplify

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

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

General solutions for

cos (a) = - \frac{1}{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}

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

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

Combine all the solutions

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

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

sin (a) - sin (2 a) = 0

Rewrite using trig identities

- sin (2 a) + sin (a)

Use the Sum to Product identity:

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

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

Simplify

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

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

\frac{a - 2 a}{2} = - \frac{a}{2}

\frac{a - 2 a}{2}

Add similar elements:

a - 2 a = - a

= \frac{- a}{2}

Apply the fraction rule:

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

= - \frac{a}{2}

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

Use the negative angle identity:

sin (- x) = - sin (x)

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

Remove parentheses:

(- a) = - a

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

Add similar elements:

a + 2 a = 3 a

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

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

- 2 cos (\frac{3 a}{2}) sin (\frac{a}{2}) = 0

Solving each part separately

cos (\frac{3 a}{2}) = 0 or sin (\frac{a}{2}) = 0

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

cos (\frac{3 a}{2}) = 0

General solutions for

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

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

Solve

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

\frac{3 a}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{3 a}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 \cdot 3 a}{2} : 3 a

\frac{2 \cdot 3 a}{2}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6 a}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 a

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

3 a = π + 4 πn

3 a = π + 4 πn

3 a = π + 4 πn

Divide both sides by

3

3 a = π + 4 πn

Divide both sides by

3

\frac{3 a}{3} = \frac{π}{3} + \frac{4 πn}{3}

Simplify

a = \frac{π}{3} + \frac{4 πn}{3}

a = \frac{π}{3} + \frac{4 πn}{3}

Solve

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

\frac{3 a}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{3 a}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 \cdot 3 a}{2} : 3 a

\frac{2 \cdot 3 a}{2}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6 a}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 a

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

3 a = 3 π + 4 πn

3 a = 3 π + 4 πn

3 a = 3 π + 4 πn

Divide both sides by

3

3 a = 3 π + 4 πn

Divide both sides by

3

\frac{3 a}{3} = \frac{3 π}{3} + \frac{4 πn}{3}

Simplify

a = π + \frac{4 πn}{3}

a = π + \frac{4 πn}{3}

a = \frac{π}{3} + \frac{4 πn}{3}, a = π + \frac{4 πn}{3}

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

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

General solutions for

sin (\frac{a}{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{a}{2} = 0 + 2 πn, \frac{a}{2} = π + 2 πn

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

Solve

\frac{a}{2} = 0 + 2 πn : a = 4 πn

\frac{a}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{a}{2} = 2 πn

Multiply both sides by

2

\frac{a}{2} = 2 πn

Multiply both sides by

2

\frac{2 a}{2} = 2 \cdot 2 πn

Simplify

a = 4 πn

a = 4 πn

Solve

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

\frac{a}{2} = π + 2 πn

Multiply both sides by

2

\frac{a}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 a}{2} = 2 π + 2 \cdot 2 πn

Simplify

a = 2 π + 4 πn

a = 2 π + 4 πn

a = 4 πn, a = 2 π + 4 πn

Combine all the solutions

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

Combine all the solutions

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

Merge Overlapping Intervals

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

Graph

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