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Popular Trigonometry >

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

Solution

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

Solution

x = \frac{3 π}{2} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Degrees

x = 27 0^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

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

Expand

2 (1 - sin^{2} (x)) : 2 - 2 sin^{2} (x)

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

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 1, c = sin^{2} (x)

= 2 \cdot 1 - 2 sin^{2} (x)

Multiply the numbers:

2 \cdot 1 = 2

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

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

Expand

2 sin (x) (1 - sin^{2} (x)) : 2 sin (x) - 2 sin^{3} (x)

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

Apply the distributive law:

a (b - c) = ab - a c

a = 2 sin (x), b = 1, c = sin^{2} (x)

= 2 sin (x) \cdot 1 - 2 sin (x) sin^{2} (x)

= 2 \cdot 1 \cdot sin (x) - 2 sin^{2} (x) sin (x)

Simplify

2 \cdot 1 \cdot sin (x) - 2 sin^{2} (x) sin (x) : 2 sin (x) - 2 sin^{3} (x)

2 \cdot 1 \cdot sin (x) - 2 sin^{2} (x) sin (x)

2 \cdot 1 \cdot sin (x) = 2 sin (x)

2 \cdot 1 \cdot sin (x)

Multiply the numbers:

2 \cdot 1 = 2

= 2 sin (x)

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

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

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

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

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

Add the numbers:

2 + 1 = 3

= 2 sin^{3} (x)

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

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

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

Simplify

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

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

Group like terms

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

Add similar elements:

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

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

Add/Subtract the numbers:

- 1 + 2 = 1

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

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

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

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

Solve by substitution

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

Let:

sin (x) = u

1 + u - 2 u^{2} - 2 u^{3} = 0

1 + u - 2 u^{2} - 2 u^{3} = 0 : u = - 1, u = - \frac{2}{2}, u = \frac{2}{2}

1 + u - 2 u^{2} - 2 u^{3} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

- 2 u^{3} - 2 u^{2} + u + 1 = 0

Factor

- 2 u^{3} - 2 u^{2} + u + 1 : - (u + 1) (2 u + 1) (2 u - 1)

- 2 u^{3} - 2 u^{2} + u + 1

Factor out common term

- 1

= - (2 u^{3} + 2 u^{2} - u - 1)

Factor

2 u^{3} + 2 u^{2} - u - 1 : (u + 1) (2 u + 1) (2 u - 1)

2 u^{3} + 2 u^{2} - u - 1

= (2 u^{3} + 2 u^{2}) + (- u - 1)

Factor out

- 1

from

- u - 1 : - (u + 1)

- u - 1

Factor out common term

- 1

= - (u + 1)

Factor out

2 u^{2}

from

2 u^{3} + 2 u^{2} : 2 u^{2} (u + 1)

2 u^{3} + 2 u^{2}

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

u^{3} = u u^{2}

= 2 u u^{2} + 2 u^{2}

Factor out common term

2 u^{2}

= 2 u^{2} (u + 1)

= - (u + 1) + 2 u^{2} (u + 1)

Factor out common term

u + 1

= (u + 1) (2 u^{2} - 1)

Factor

2 u^{2} - 1 : (2 u + 1) (2 u - 1)

2 u^{2} - 1

Rewrite

2 u^{2} - 1

(2 u)^{2} - 1^{2}

2 u^{2} - 1

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

= (2)^{2} u^{2} - 1

Rewrite

1

1^{2}

= (2)^{2} u^{2} - 1^{2}

Apply exponent rule:

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

(2)^{2} u^{2} = (2 u)^{2}

= (2 u)^{2} - 1^{2}

= (2 u)^{2} - 1^{2}

Apply Difference of Two Squares Formula:

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

(2 u)^{2} - 1^{2} = (2 u + 1) (2 u - 1)

= (2 u + 1) (2 u - 1)

= (u + 1) (2 u + 1) (2 u - 1)

= - (u + 1) (2 u + 1) (2 u - 1)

- (u + 1) (2 u + 1) (2 u - 1) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u + 1 = 0 or 2 u + 1 = 0 or 2 u - 1 = 0

Solve

u + 1 = 0 : u = - 1

u + 1 = 0

Move

1

to the right side

u + 1 = 0

Subtract

1

from both sides

u + 1 - 1 = 0 - 1

Simplify

u = - 1

u = - 1

Solve

2 u + 1 = 0 : u = - \frac{2}{2}

2 u + 1 = 0

Move

1

to the right side

2 u + 1 = 0

Subtract

1

from both sides

2 u + 1 - 1 = 0 - 1

Simplify

2 u = - 1

2 u = - 1

Divide both sides by

2

2 u = - 1

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

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}

u = - \frac{2}{2}

u = - \frac{2}{2}

u = - \frac{2}{2}

Solve

2 u - 1 = 0 : u = \frac{2}{2}

2 u - 1 = 0

Move

1

to the right side

2 u - 1 = 0

Add

1

to both sides

2 u - 1 + 1 = 0 + 1

Simplify

2 u = 1

2 u = 1

Divide both sides by

2

2 u = 1

Divide both sides by

2

\frac{2 u}{2} = \frac{1}{2}

Simplify

\frac{2 u}{2} = \frac{1}{2}

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\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}

u = \frac{2}{2}

u = \frac{2}{2}

u = \frac{2}{2}

The solutions are

u = - 1, u = - \frac{2}{2}, u = \frac{2}{2}

Substitute back

u = sin (x)

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

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

sin (x) = - 1 : x = \frac{3 π}{2} + 2 πn

sin (x) = - 1

General solutions for

sin (x) = - 1

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}

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

sin (x) = - \frac{2}{2} : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

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

General solutions for

sin (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}

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

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

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

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

General solutions for

sin (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}

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

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

Combine all the solutions

x = \frac{3 π}{2} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Graph

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