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Popular Trigonometry >

1-8sin^2(x)cos^2(x)=0

Solution

1 - 8 sin^{2} (x) cos^{2} (x) = 0

Solution

x = \frac{5 π}{8} + πn, x = \frac{7 π}{8} + πn, x = \frac{π}{8} + πn, x = \frac{3 π}{8} + πn

Degrees

x = 112. 5^{\circ} + 18 0^{\circ} n, x = 157. 5^{\circ} + 18 0^{\circ} n, x = 22. 5^{\circ} + 18 0^{\circ} n, x = 67. 5^{\circ} + 18 0^{\circ} n

Solution steps

1 - 8 sin^{2} (x) cos^{2} (x) = 0

Factor

1 - 8 sin^{2} (x) cos^{2} (x) : (22 sin (x) cos (x) + 1) (- 22 sin (x) cos (x) + 1)

1 - 8 sin^{2} (x) cos^{2} (x)

Rewrite

1 - 8 sin^{2} (x) cos^{2} (x)

1 - (8 sin (x) cos (x))^{2}

1 - 8 sin^{2} (x) cos^{2} (x)

Apply radical rule:

a = (a)^{2}

8 = (8)^{2}

= 1 - (8)^{2} sin^{2} (x) cos^{2} (x)

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

(8)^{2} sin^{2} (x) cos^{2} (x) = (8 sin (x) cos (x))^{2}

= 1 - (8 sin (x) cos (x))^{2}

= 1 - (8 sin (x) cos (x))^{2}

Apply Difference of Two Squares Formula:

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

1 - (8 sin (x) cos (x))^{2} = (8 sin (x) cos (x) + 1) (- 8 sin (x) cos (x) + 1)

= (8 sin (x) cos (x) + 1) (- 8 sin (x) cos (x) + 1)

Refine

= (22 sin (x) cos (x) + 1) (- 22 sin (x) cos (x) + 1)

(22 sin (x) cos (x) + 1) (- 22 sin (x) cos (x) + 1) = 0

Solving each part separately

22 sin (x) cos (x) + 1 = 0 or - 22 sin (x) cos (x) + 1 = 0

22 sin (x) cos (x) + 1 = 0 : x = \frac{5 π}{8} + πn, x = \frac{7 π}{8} + πn

22 sin (x) cos (x) + 1 = 0

Rewrite using trig identities

22 sin (x) cos (x) + 1

Use the Double Angle identity:

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

= 1 + sin (2 x) 2

1 + sin (2 x) 2 = 0

Move

1

to the right side

1 + sin (2 x) 2 = 0

Subtract

1

from both sides

1 + sin (2 x) 2 - 1 = 0 - 1

Simplify

sin (2 x) 2 = - 1

sin (2 x) 2 = - 1

Divide both sides by

2

sin (2 x) 2 = - 1

Divide both sides by

2

\frac{sin ( 2 x ) 2}{2} = \frac{- 1}{2}

Simplify

\frac{sin ( 2 x ) 2}{2} = \frac{- 1}{2}

Simplify

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

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

Cancel the common factor:

2

= sin (2 x)

Simplify

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

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

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

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

General solutions for

sin (2 x) = - \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}

2 x = \frac{5 π}{4} + 2 πn, 2 x = \frac{7 π}{4} + 2 πn

2 x = \frac{5 π}{4} + 2 πn, 2 x = \frac{7 π}{4} + 2 πn

Solve

2 x = \frac{5 π}{4} + 2 πn : x = \frac{5 π}{8} + πn

2 x = \frac{5 π}{4} + 2 πn

Divide both sides by

2

2 x = \frac{5 π}{4} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{5 π}{4}}{2} + \frac{2 πn}{2} : \frac{5 π}{8} + πn

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

\frac{\frac{5 π}{4}}{2} = \frac{5 π}{8}

\frac{\frac{5 π}{4}}{2}

Apply the fraction rule:

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

= \frac{5 π}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{5 π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{5 π}{8} + πn

x = \frac{5 π}{8} + πn

x = \frac{5 π}{8} + πn

x = \frac{5 π}{8} + πn

Solve

2 x = \frac{7 π}{4} + 2 πn : x = \frac{7 π}{8} + πn

2 x = \frac{7 π}{4} + 2 πn

Divide both sides by

2

2 x = \frac{7 π}{4} + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2} : \frac{7 π}{8} + πn

\frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{7 π}{4}}{2} = \frac{7 π}{8}

\frac{\frac{7 π}{4}}{2}

Apply the fraction rule:

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

= \frac{7 π}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{7 π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{7 π}{8} + πn

x = \frac{7 π}{8} + πn

x = \frac{7 π}{8} + πn

x = \frac{7 π}{8} + πn

x = \frac{5 π}{8} + πn, x = \frac{7 π}{8} + πn

- 22 sin (x) cos (x) + 1 = 0 : x = \frac{π}{8} + πn, x = \frac{3 π}{8} + πn

- 22 sin (x) cos (x) + 1 = 0

Rewrite using trig identities

- 22 sin (x) cos (x) + 1

Use the Double Angle identity:

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

= 1 - sin (2 x) 2

1 - sin (2 x) 2 = 0

Move

1

to the right side

1 - sin (2 x) 2 = 0

Subtract

1

from both sides

1 - sin (2 x) 2 - 1 = 0 - 1

Simplify

- sin (2 x) 2 = - 1

- sin (2 x) 2 = - 1

Divide both sides by

- 2

- sin (2 x) 2 = - 1

Divide both sides by

- 2

\frac{- sin ( 2 x ) 2}{- 2} = \frac{- 1}{- 2}

Simplify

\frac{- sin ( 2 x ) 2}{- 2} = \frac{- 1}{- 2}

Simplify

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

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

Apply the fraction rule:

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

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

Cancel the common factor:

2

= sin (2 x)

Simplify

\frac{- 1}{- 2} : \frac{2}{2}

\frac{- 1}{- 2}

Apply the fraction rule:

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

= \frac{1}{2}

Rationalize

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

= \frac{2}{2}

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

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

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

General solutions for

sin (2 x) = \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}

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

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

Solve

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

2 x = \frac{π}{4} + 2 πn

Divide both sides by

2

2 x = \frac{π}{4} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{π}{4}}{2} + \frac{2 πn}{2} : \frac{π}{8} + πn

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

\frac{\frac{π}{4}}{2} = \frac{π}{8}

\frac{\frac{π}{4}}{2}

Apply the fraction rule:

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

= \frac{π}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{8} + πn

x = \frac{π}{8} + πn

x = \frac{π}{8} + πn

x = \frac{π}{8} + πn

Solve

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

2 x = \frac{3 π}{4} + 2 πn

Divide both sides by

2

2 x = \frac{3 π}{4} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

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

\frac{\frac{3 π}{4}}{2} = \frac{3 π}{8}

\frac{\frac{3 π}{4}}{2}

Apply the fraction rule:

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

= \frac{3 π}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{3 π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{3 π}{8} + πn

x = \frac{3 π}{8} + πn

x = \frac{3 π}{8} + πn

x = \frac{3 π}{8} + πn

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

Combine all the solutions

x = \frac{5 π}{8} + πn, x = \frac{7 π}{8} + πn, x = \frac{π}{8} + πn, x = \frac{3 π}{8} + πn

Graph

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