What is the general solution for 2sin(pi/4+x)*sin(pi/4-x)+sin^2(x)=0 ?

The general solution for 2sin(pi/4+x)*sin(pi/4-x)+sin^2(x)=0 is x= pi/2+2pin,x=(3pi)/2+2pin

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2sin(pi/4+x)*sin(pi/4-x)+sin^2(x)=0

Solution

2 sin (\frac{π}{4} + x) \cdot sin (\frac{π}{4} - x) + sin^{2} (x) = 0

Solution

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

Degrees

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

Solution steps

2 sin (\frac{π}{4} + x) sin (\frac{π}{4} - x) + sin^{2} (x) = 0

Rewrite using trig identities

2 sin (\frac{π}{4} + x) sin (\frac{π}{4} - x) + sin^{2} (x) = 0

Rewrite using trig identities

sin (\frac{π}{4} - x)

Use the Angle Difference identity:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= sin (\frac{π}{4}) cos (x) - cos (\frac{π}{4}) sin (x)

Simplify

sin (\frac{π}{4}) cos (x) - cos (\frac{π}{4}) sin (x) : \frac{2 cos ( x ) - 2 sin ( x )}{2}

sin (\frac{π}{4}) cos (x) - cos (\frac{π}{4}) sin (x)

sin (\frac{π}{4}) cos (x) = \frac{2 cos ( x )}{2}

sin (\frac{π}{4}) cos (x)

Simplify

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{2}

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{2}{2}

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

Multiply fractions:

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

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

cos (\frac{π}{4}) sin (x) = \frac{2 sin ( x )}{2}

cos (\frac{π}{4}) sin (x)

Simplify

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

Use the following trivial identity:

cos (\frac{π}{4}) = \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}

= \frac{2}{2}

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

Multiply fractions:

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

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

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

Apply rule

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

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

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

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= sin (\frac{π}{4}) cos (x) + cos (\frac{π}{4}) sin (x)

Simplify

sin (\frac{π}{4}) cos (x) + cos (\frac{π}{4}) sin (x) : \frac{2 cos ( x ) + 2 sin ( x )}{2}

sin (\frac{π}{4}) cos (x) + cos (\frac{π}{4}) sin (x)

sin (\frac{π}{4}) cos (x) = \frac{2 cos ( x )}{2}

sin (\frac{π}{4}) cos (x)

Simplify

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{2}

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{2}{2}

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

Multiply fractions:

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

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

cos (\frac{π}{4}) sin (x) = \frac{2 sin ( x )}{2}

cos (\frac{π}{4}) sin (x)

Simplify

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

Use the following trivial identity:

cos (\frac{π}{4}) = \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}

= \frac{2}{2}

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

Multiply fractions:

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

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

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

Apply rule

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

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

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

2 \cdot \frac{2 cos ( x ) + 2 sin ( x )}{2} \cdot \frac{2 cos ( x ) - 2 sin ( x )}{2} + sin^{2} (x) = 0

Simplify

2 \cdot \frac{2 cos ( x ) + 2 sin ( x )}{2} \cdot \frac{2 cos ( x ) - 2 sin ( x )}{2} + sin^{2} (x) : cos^{2} (x)

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

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

Factor

(2 cos (x) + 2 sin (x)) (2 cos (x) - 2 sin (x)) : 2 (cos (x) + sin (x)) (cos (x) - sin (x))

(2 cos (x) + 2 sin (x)) (2 cos (x) - 2 sin (x))

Factor

2 cos (x) + 2 sin (x) : 2 (cos (x) + sin (x))

2 cos (x) + 2 sin (x)

Factor out common term

2

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

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

Factor

2 cos (x) - 2 sin (x) : 2 (cos (x) - sin (x))

2 cos (x) - 2 sin (x)

Factor out common term

2

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

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

Refine

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

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

Divide the numbers:

\frac{2}{2} = 1

= (cos (x) + sin (x)) (cos (x) - sin (x))

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

Expand

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

(cos (x) + sin (x)) (cos (x) - sin (x))

Apply Difference of Two Squares Formula:

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

a = cos (x), b = sin (x)

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

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

Add similar elements:

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

= cos^{2} (x)

cos^{2} (x) = 0

cos^{2} (x) = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

cos (x) = 0

General solutions for

cos (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}

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

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

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 2sin(pi/4+x)*sin(pi/4-x)+sin^2(x)=0 ?
The general solution for 2sin(pi/4+x)*sin(pi/4-x)+sin^2(x)=0 is x= pi/2+2pin,x=(3pi)/2+2pin