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Popular Trigonometry >

-2sin^2(3x)=cos(3x)-2

Solution

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

Solution

x = \frac{π}{9} + \frac{2 πn}{3}, x = \frac{5 π}{9} + \frac{2 πn}{3}, x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}

Degrees

x = 2 0^{\circ} + 12 0^{\circ} n, x = 10 0^{\circ} + 12 0^{\circ} n, x = 3 0^{\circ} + 12 0^{\circ} n, x = 9 0^{\circ} + 12 0^{\circ} n

Solution steps

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

Subtract

cos (3 x) - 2

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

a = - 2, b = 1, c = cos^{2} (3 x)

= - 2 \cdot 1 - (- 2) cos^{2} (3 x)

Apply minus-plus rules

- (- a) = a

= - 2 \cdot 1 + 2 cos^{2} (3 x)

Multiply the numbers:

2 \cdot 1 = 2

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

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

Simplify

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

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

Group like terms

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

2 - 2 = 0

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

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

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

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

Solve by substitution

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

Let:

cos (3 x) = u

- u + 2 u^{2} = 0

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

- u + 2 u^{2} = 0

Write in the standard form

a x^{2} + b x + c = 0

2 u^{2} - u = 0

Solve with the quadratic formula

2 u^{2} - u = 0

Quadratic Equation Formula:

For

a = 2, b = - 1, c = 0

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 2 \cdot 0}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 2 \cdot 0}{2 \cdot 2}

(- 1)^{2} - 4 \cdot 2 \cdot 0 = 1

(- 1)^{2} - 4 \cdot 2 \cdot 0

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

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

= 1^{2}

Apply rule

1^{a} = 1

= 1

4 \cdot 2 \cdot 0 = 0

4 \cdot 2 \cdot 0

Apply rule

0 \cdot a = 0

= 0

= 1 - 0

Subtract the numbers:

1 - 0 = 1

= 1

Apply rule

1 = 1

= 1

u_{1, 2} = \frac{- ( - 1 ) \pm 1}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- ( - 1 ) + 1}{2 \cdot 2}, u_{2} = \frac{- ( - 1 ) - 1}{2 \cdot 2}

u = \frac{- ( - 1 ) + 1}{2 \cdot 2} : \frac{1}{2}

\frac{- ( - 1 ) + 1}{2 \cdot 2}

Apply rule

- (- a) = a

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

Add the numbers:

1 + 1 = 2

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

u = \frac{- ( - 1 ) - 1}{2 \cdot 2} : 0

\frac{- ( - 1 ) - 1}{2 \cdot 2}

Apply rule

- (- a) = a

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

Subtract the numbers:

1 - 1 = 0

= \frac{0}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{0}{4}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

The solutions to the quadratic equation are:

u = \frac{1}{2}, u = 0

Substitute back

u = cos (3 x)

cos (3 x) = \frac{1}{2}, cos (3 x) = 0

cos (3 x) = \frac{1}{2}, cos (3 x) = 0

cos (3 x) = \frac{1}{2} : x = \frac{π}{9} + \frac{2 πn}{3}, x = \frac{5 π}{9} + \frac{2 πn}{3}

cos (3 x) = \frac{1}{2}

General solutions for

cos (3 x) = \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}

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

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

Solve

3 x = \frac{π}{3} + 2 πn : x = \frac{π}{9} + \frac{2 πn}{3}

3 x = \frac{π}{3} + 2 πn

Divide both sides by

3

3 x = \frac{π}{3} + 2 πn

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{\frac{π}{3}}{3} + \frac{2 πn}{3} : \frac{π}{9} + \frac{2 πn}{3}

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

\frac{\frac{π}{3}}{3} = \frac{π}{9}

\frac{\frac{π}{3}}{3}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{π}{9}

= \frac{π}{9} + \frac{2 πn}{3}

x = \frac{π}{9} + \frac{2 πn}{3}

x = \frac{π}{9} + \frac{2 πn}{3}

x = \frac{π}{9} + \frac{2 πn}{3}

Solve

3 x = \frac{5 π}{3} + 2 πn : x = \frac{5 π}{9} + \frac{2 πn}{3}

3 x = \frac{5 π}{3} + 2 πn

Divide both sides by

3

3 x = \frac{5 π}{3} + 2 πn

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{\frac{5 π}{3}}{3} + \frac{2 πn}{3} : \frac{5 π}{9} + \frac{2 πn}{3}

\frac{\frac{5 π}{3}}{3} + \frac{2 πn}{3}

\frac{\frac{5 π}{3}}{3} = \frac{5 π}{9}

\frac{\frac{5 π}{3}}{3}

Apply the fraction rule:

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

= \frac{5 π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{5 π}{9}

= \frac{5 π}{9} + \frac{2 πn}{3}

x = \frac{5 π}{9} + \frac{2 πn}{3}

x = \frac{5 π}{9} + \frac{2 πn}{3}

x = \frac{5 π}{9} + \frac{2 πn}{3}

x = \frac{π}{9} + \frac{2 πn}{3}, x = \frac{5 π}{9} + \frac{2 πn}{3}

cos (3 x) = 0 : x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}

cos (3 x) = 0

General solutions for

cos (3 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}

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

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

Solve

3 x = \frac{π}{2} + 2 πn : x = \frac{π}{6} + \frac{2 πn}{3}

3 x = \frac{π}{2} + 2 πn

Divide both sides by

3

3 x = \frac{π}{2} + 2 πn

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{\frac{π}{2}}{3} + \frac{2 πn}{3} : \frac{π}{6} + \frac{2 πn}{3}

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

\frac{\frac{π}{2}}{3} = \frac{π}{6}

\frac{\frac{π}{2}}{3}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{π}{6}

= \frac{π}{6} + \frac{2 πn}{3}

x = \frac{π}{6} + \frac{2 πn}{3}

x = \frac{π}{6} + \frac{2 πn}{3}

x = \frac{π}{6} + \frac{2 πn}{3}

Solve

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

3 x = \frac{3 π}{2} + 2 πn

Divide both sides by

3

3 x = \frac{3 π}{2} + 2 πn

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

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

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

\frac{\frac{3 π}{2}}{3} = \frac{π}{2}

\frac{\frac{3 π}{2}}{3}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{3 π}{6}

Cancel the common factor:

3

= \frac{π}{2}

= \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}

Combine all the solutions

x = \frac{π}{9} + \frac{2 πn}{3}, x = \frac{5 π}{9} + \frac{2 πn}{3}, x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}

Graph

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