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Popular >

2cos(2x)-sqrt(3)cos(x)=0

Solution

2 cos (2 x) - 3 cos (x) = 0

Solution

x = 0.29769 \dots + 2 πn, x = 2 π - 0.29769 \dots + 2 πn, x = 2.12116 \dots + 2 πn, x = - 2.12116 \dots + 2 πn

Degrees

x = 17.05643 \dots^{\circ} + 36 0^{\circ} n, x = 342.94356 \dots^{\circ} + 36 0^{\circ} n, x = 121.53394 \dots^{\circ} + 36 0^{\circ} n, x = - 121.53394 \dots^{\circ} + 36 0^{\circ} n

Solution steps

2 cos (2 x) - 3 cos (x) = 0

Rewrite using trig identities

2 cos (2 x) - cos (x) 3

Use the Double Angle identity:

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

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

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

Solve by substitution

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

Let:

cos (x) = u

(- 1 + 2 u^{2}) \cdot 2 - u 3 = 0

(- 1 + 2 u^{2}) \cdot 2 - u 3 = 0 : u = \frac{3 + 35}{8}, u = \frac{3 - 35}{8}

(- 1 + 2 u^{2}) \cdot 2 - u 3 = 0

Expand

(- 1 + 2 u^{2}) \cdot 2 - u 3 : - 2 + 4 u^{2} - 3 u

(- 1 + 2 u^{2}) \cdot 2 - u 3

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

Expand

2 (- 1 + 2 u^{2}) : - 2 + 4 u^{2}

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

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = - 1, c = 2 u^{2}

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

Apply minus-plus rules

+ (- a) = - a

= - 2 \cdot 1 + 2 \cdot 2 u^{2}

Simplify

- 2 \cdot 1 + 2 \cdot 2 u^{2} : - 2 + 4 u^{2}

- 2 \cdot 1 + 2 \cdot 2 u^{2}

Multiply the numbers:

2 \cdot 1 = 2

= - 2 + 2 \cdot 2 u^{2}

Multiply the numbers:

2 \cdot 2 = 4

= - 2 + 4 u^{2}

= - 2 + 4 u^{2}

= - 2 + 4 u^{2} - u 3

= - 2 + 4 u^{2} - 3 u

- 2 + 4 u^{2} - 3 u = 0

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = 4, b = - 3, c = - 2

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

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

(- 3)^{2} - 4 \cdot 4 (- 2) = 35

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

Apply rule

- (- a) = a

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

(- 3)^{2} = 3

(- 3)^{2}

Apply exponent rule:

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

n

is even

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

= (3)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 3

4 \cdot 4 \cdot 2 = 32

4 \cdot 4 \cdot 2

Multiply the numbers:

4 \cdot 4 \cdot 2 = 32

= 32

= 3 + 32

Add the numbers:

3 + 32 = 35

= 35

u_{1, 2} = \frac{- ( - 3 ) \pm 35}{2 \cdot 4}

Separate the solutions

u_{1} = \frac{- ( - 3 ) + 35}{2 \cdot 4}, u_{2} = \frac{- ( - 3 ) - 35}{2 \cdot 4}

u = \frac{- ( - 3 ) + 35}{2 \cdot 4} : \frac{3 + 35}{8}

\frac{- ( - 3 ) + 35}{2 \cdot 4}

Apply rule

- (- a) = a

= \frac{3 + 35}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{3 + 35}{8}

u = \frac{- ( - 3 ) - 35}{2 \cdot 4} : \frac{3 - 35}{8}

\frac{- ( - 3 ) - 35}{2 \cdot 4}

Apply rule

- (- a) = a

= \frac{3 - 35}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{3 - 35}{8}

The solutions to the quadratic equation are:

u = \frac{3 + 35}{8}, u = \frac{3 - 35}{8}

Substitute back

u = cos (x)

cos (x) = \frac{3 + 35}{8}, cos (x) = \frac{3 - 35}{8}

cos (x) = \frac{3 + 35}{8}, cos (x) = \frac{3 - 35}{8}

cos (x) = \frac{3 + 35}{8} : x = arccos (\frac{3 + 35}{8}) + 2 πn, x = 2 π - arccos (\frac{3 + 35}{8}) + 2 πn

cos (x) = \frac{3 + 35}{8}

Apply trig inverse properties

cos (x) = \frac{3 + 35}{8}

General solutions for

cos (x) = \frac{3 + 35}{8}

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

x = arccos (\frac{3 + 35}{8}) + 2 πn, x = 2 π - arccos (\frac{3 + 35}{8}) + 2 πn

x = arccos (\frac{3 + 35}{8}) + 2 πn, x = 2 π - arccos (\frac{3 + 35}{8}) + 2 πn

cos (x) = \frac{3 - 35}{8} : x = arccos (\frac{3 - 35}{8}) + 2 πn, x = - arccos (\frac{3 - 35}{8}) + 2 πn

cos (x) = \frac{3 - 35}{8}

Apply trig inverse properties

cos (x) = \frac{3 - 35}{8}

General solutions for

cos (x) = \frac{3 - 35}{8}

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

x = arccos (\frac{3 - 35}{8}) + 2 πn, x = - arccos (\frac{3 - 35}{8}) + 2 πn

x = arccos (\frac{3 - 35}{8}) + 2 πn, x = - arccos (\frac{3 - 35}{8}) + 2 πn

Combine all the solutions

x = arccos (\frac{3 + 35}{8}) + 2 πn, x = 2 π - arccos (\frac{3 + 35}{8}) + 2 πn, x = arccos (\frac{3 - 35}{8}) + 2 πn, x = - arccos (\frac{3 - 35}{8}) + 2 πn

Show solutions in decimal form

x = 0.29769 \dots + 2 πn, x = 2 π - 0.29769 \dots + 2 πn, x = 2.12116 \dots + 2 πn, x = - 2.12116 \dots + 2 πn

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