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Popular >

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

Solution

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

Solution

x = \frac{0.81462 \dots}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{0.81462 \dots}{3} + \frac{2 πn}{3}

Degrees

x = 15.55820 \dots^{\circ} + 12 0^{\circ} n, x = 104.44179 \dots^{\circ} + 12 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Multiply the numbers:

2 \cdot 1 = 2

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

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

Add the numbers:

1 + 2 = 3

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

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

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

Solve by substitution

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

Let:

cos (3 x) = u

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

3 - 2 u^{2} - 3 u = 0 : u = - \frac{3 + 33}{4}, u = \frac{33 - 3}{4}

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = - 2, b = - 3, c = 3

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 ( - 2 ) \cdot 3}{2 ( - 2 )}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 ( - 2 ) \cdot 3}{2 ( - 2 )}

(- 3)^{2} - 4 (- 2) \cdot 3 = 33

(- 3)^{2} - 4 (- 2) \cdot 3

Apply rule

- (- a) = a

= (- 3)^{2} + 4 \cdot 2 \cdot 3

Apply exponent rule:

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

n

is even

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

= 3^{2} + 4 \cdot 2 \cdot 3

Multiply the numbers:

4 \cdot 2 \cdot 3 = 24

= 3^{2} + 24

3^{2} = 9

= 9 + 24

Add the numbers:

9 + 24 = 33

= 33

u_{1, 2} = \frac{- ( - 3 ) \pm 33}{2 ( - 2 )}

Separate the solutions

u_{1} = \frac{- ( - 3 ) + 33}{2 ( - 2 )}, u_{2} = \frac{- ( - 3 ) - 33}{2 ( - 2 )}

u = \frac{- ( - 3 ) + 33}{2 ( - 2 )} : - \frac{3 + 33}{4}

\frac{- ( - 3 ) + 33}{2 ( - 2 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{3 + 33}{- 2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 + 33}{- 4}

Apply the fraction rule:

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

= - \frac{3 + 33}{4}

u = \frac{- ( - 3 ) - 33}{2 ( - 2 )} : \frac{33 - 3}{4}

\frac{- ( - 3 ) - 33}{2 ( - 2 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{3 - 33}{- 2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 - 33}{- 4}

Apply the fraction rule:

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

3 - 33 = - (33 - 3)

= \frac{33 - 3}{4}

The solutions to the quadratic equation are:

u = - \frac{3 + 33}{4}, u = \frac{33 - 3}{4}

Substitute back

u = cos (3 x)

cos (3 x) = - \frac{3 + 33}{4}, cos (3 x) = \frac{33 - 3}{4}

cos (3 x) = - \frac{3 + 33}{4}, cos (3 x) = \frac{33 - 3}{4}

cos (3 x) = - \frac{3 + 33}{4} :

No Solution

cos (3 x) = - \frac{3 + 33}{4}

- 1 \leq cos (x) \leq 1

No Solution

cos (3 x) = \frac{33 - 3}{4} : x = \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}

cos (3 x) = \frac{33 - 3}{4}

Apply trig inverse properties

cos (3 x) = \frac{33 - 3}{4}

General solutions for

cos (3 x) = \frac{33 - 3}{4}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

3 x = arccos (\frac{33 - 3}{4}) + 2 πn, 3 x = 2 π - arccos (\frac{33 - 3}{4}) + 2 πn

3 x = arccos (\frac{33 - 3}{4}) + 2 πn, 3 x = 2 π - arccos (\frac{33 - 3}{4}) + 2 πn

Solve

3 x = arccos (\frac{33 - 3}{4}) + 2 πn : x = \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}

3 x = arccos (\frac{33 - 3}{4}) + 2 πn

Divide both sides by

3

3 x = arccos (\frac{33 - 3}{4}) + 2 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}

Simplify

x = \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}

x = \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}

Solve

3 x = 2 π - arccos (\frac{33 - 3}{4}) + 2 πn : x = \frac{2 π}{3} - \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}

3 x = 2 π - arccos (\frac{33 - 3}{4}) + 2 πn

Divide both sides by

3

3 x = 2 π - arccos (\frac{33 - 3}{4}) + 2 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{2 π}{3} - \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}

Simplify

x = \frac{2 π}{3} - \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}

x = \frac{2 π}{3} - \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}

x = \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}

Combine all the solutions

x = \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{arccos ( \frac{33 - 3}{4} )}{3} + \frac{2 πn}{3}

Show solutions in decimal form

x = \frac{0.81462 \dots}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{0.81462 \dots}{3} + \frac{2 πn}{3}

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