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Popular Trigonometry >

4cos(3x)-1=2sin(3x)tan(3x)

Solution

4 cos (3 x) - 1 = 2 sin (3 x) tan (3 x)

Solution

x = \frac{0.84106 \dots}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{0.84106 \dots}{3} + \frac{2 πn}{3}, x = \frac{2 π}{9} + \frac{2 πn}{3}, x = \frac{4 π}{9} + \frac{2 πn}{3}

Degrees

x = 16.06322 \dots^{\circ} + 12 0^{\circ} n, x = 103.93677 \dots^{\circ} + 12 0^{\circ} n, x = 4 0^{\circ} + 12 0^{\circ} n, x = 8 0^{\circ} + 12 0^{\circ} n

Solution steps

4 cos (3 x) - 1 = 2 sin (3 x) tan (3 x)

Subtract

2 sin (3 x) tan (3 x)

from both sides

4 cos (3 x) - 1 - 2 sin (3 x) tan (3 x) = 0

Rewrite using trig identities

- 1 + 4 cos (3 x) - 2 sin (3 x) tan (3 x)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= - 1 + 4 cos (3 x) - 2 sin (3 x) \frac{sin ( 3 x )}{cos ( 3 x )}

2 sin (3 x) \frac{sin ( 3 x )}{cos ( 3 x )} = \frac{2 sin ^{2} ( 3 x )}{cos ( 3 x )}

2 sin (3 x) \frac{sin ( 3 x )}{cos ( 3 x )}

Multiply fractions:

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

= \frac{sin ( 3 x ) \cdot 2 sin ( 3 x )}{cos ( 3 x )}

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

sin (3 x) \cdot 2 sin (3 x)

Apply exponent rule:

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

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

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

Add the numbers:

1 + 1 = 2

= 2 sin^{2} (3 x)

= \frac{2 sin ^{2} ( 3 x )}{cos ( 3 x )}

= - 1 + 4 cos (3 x) - \frac{2 sin ^{2} ( 3 x )}{cos ( 3 x )}

Use the Pythagorean identity:

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

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

= - 1 - \frac{2 ( 1 - cos ^{2} ( 3 x ) )}{cos ( 3 x )} + 4 cos (3 x)

Combine the fractions

- \frac{2 ( - cos ^{2} ( 3 x ) + 1 )}{cos ( 3 x )} + 4 cos (3 x) : \frac{- 2 + 6 cos ^{2} ( 3 x )}{cos ( 3 x )}

- \frac{2 ( - cos ^{2} ( 3 x ) + 1 )}{cos ( 3 x )} + 4 cos (3 x)

Convert element to fraction:

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

= - \frac{2 ( 1 - cos ^{2} ( 3 x ) )}{cos ( 3 x )} + \frac{4 cos ( 3 x ) cos ( 3 x )}{cos ( 3 x )}

Since the denominators are equal, combine the fractions:

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

= \frac{- 2 ( 1 - cos ^{2} ( 3 x ) ) + 4 cos ( 3 x ) cos ( 3 x )}{cos ( 3 x )}

- 2 (1 - cos^{2} (3 x)) + 4 cos (3 x) cos (3 x) = - 2 (1 - cos^{2} (3 x)) + 4 cos^{2} (3 x)

- 2 (1 - cos^{2} (3 x)) + 4 cos (3 x) cos (3 x)

4 cos (3 x) cos (3 x) = 4 cos^{2} (3 x)

4 cos (3 x) cos (3 x)

Apply exponent rule:

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

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

= 4 cos^{1 + 1} (3 x)

Add the numbers:

1 + 1 = 2

= 4 cos^{2} (3 x)

= - 2 (- cos^{2} (3 x) + 1) + 4 cos^{2} (3 x)

= \frac{- 2 ( - cos ^{2} ( 3 x ) + 1 ) + 4 cos ^{2} ( 3 x )}{cos ( 3 x )}

Expand

- 2 (1 - cos^{2} (3 x)) + 4 cos^{2} (3 x) : - 2 + 6 cos^{2} (3 x)

- 2 (1 - cos^{2} (3 x)) + 4 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 + 2 cos^{2} (3 x) + 4 cos^{2} (3 x)

Add similar elements:

2 cos^{2} (3 x) + 4 cos^{2} (3 x) = 6 cos^{2} (3 x)

= - 2 + 6 cos^{2} (3 x)

= \frac{- 2 + 6 cos ^{2} ( 3 x )}{cos ( 3 x )}

= \frac{- 2 + 6 cos ^{2} ( 3 x )}{cos ( 3 x )} - 1

- 1 + \frac{- 2 + 6 cos ^{2} ( 3 x )}{cos ( 3 x )} = 0

- 1 + \frac{- 2 + 6 cos ^{2} ( 3 x )}{cos ( 3 x )} = 0

Solve by substitution

- 1 + \frac{- 2 + 6 cos ^{2} ( 3 x )}{cos ( 3 x )} = 0

Let:

cos (3 x) = u

- 1 + \frac{- 2 + 6 u ^{2}}{u} = 0

- 1 + \frac{- 2 + 6 u ^{2}}{u} = 0 : u = \frac{2}{3}, u = - \frac{1}{2}

- 1 + \frac{- 2 + 6 u ^{2}}{u} = 0

Multiply both sides by

u

- 1 + \frac{- 2 + 6 u ^{2}}{u} = 0

Multiply both sides by

u

- 1 \cdot u + \frac{- 2 + 6 u ^{2}}{u} u = 0 \cdot u

Simplify

- 1 \cdot u + \frac{- 2 + 6 u ^{2}}{u} u = 0 \cdot u

Simplify

- 1 \cdot u : - u

- 1 \cdot u

Multiply:

1 \cdot u = u

= - u

Simplify

\frac{- 2 + 6 u ^{2}}{u} u : - 2 + 6 u^{2}

\frac{- 2 + 6 u ^{2}}{u} u

Multiply fractions:

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

= \frac{( - 2 + 6 u ^{2} ) u}{u}

Cancel the common factor:

u

= - - 2 + 6 u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

- u - 2 + 6 u^{2} = 0

- u - 2 + 6 u^{2} = 0

- u - 2 + 6 u^{2} = 0

Solve

- u - 2 + 6 u^{2} = 0 : u = \frac{2}{3}, u = - \frac{1}{2}

- u - 2 + 6 u^{2} = 0

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = 6, b = - 1, c = - 2

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 6 ( - 2 )}{2 \cdot 6}

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 6 ( - 2 )}{2 \cdot 6}

(- 1)^{2} - 4 \cdot 6 (- 2) = 7

(- 1)^{2} - 4 \cdot 6 (- 2)

Apply rule

- (- a) = a

= (- 1)^{2} + 4 \cdot 6 \cdot 2

(- 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 6 \cdot 2 = 48

4 \cdot 6 \cdot 2

Multiply the numbers:

4 \cdot 6 \cdot 2 = 48

= 48

= 1 + 48

Add the numbers:

1 + 48 = 49

= 49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

7^{2} = 7

= 7

u_{1, 2} = \frac{- ( - 1 ) \pm 7}{2 \cdot 6}

Separate the solutions

u_{1} = \frac{- ( - 1 ) + 7}{2 \cdot 6}, u_{2} = \frac{- ( - 1 ) - 7}{2 \cdot 6}

u = \frac{- ( - 1 ) + 7}{2 \cdot 6} : \frac{2}{3}

\frac{- ( - 1 ) + 7}{2 \cdot 6}

Apply rule

- (- a) = a

= \frac{1 + 7}{2 \cdot 6}

Add the numbers:

1 + 7 = 8

= \frac{8}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{8}{12}

Cancel the common factor:

4

= \frac{2}{3}

u = \frac{- ( - 1 ) - 7}{2 \cdot 6} : - \frac{1}{2}

\frac{- ( - 1 ) - 7}{2 \cdot 6}

Apply rule

- (- a) = a

= \frac{1 - 7}{2 \cdot 6}

Subtract the numbers:

1 - 7 = - 6

= \frac{- 6}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{- 6}{12}

Apply the fraction rule:

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

= - \frac{6}{12}

Cancel the common factor:

6

= - \frac{1}{2}

The solutions to the quadratic equation are:

u = \frac{2}{3}, u = - \frac{1}{2}

u = \frac{2}{3}, u = - \frac{1}{2}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- 1 + \frac{- 2 + 6 u ^{2}}{u}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = \frac{2}{3}, u = - \frac{1}{2}

Substitute back

u = cos (3 x)

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

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

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

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

Apply trig inverse properties

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

General solutions for

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

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

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

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

Solve

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

3 x = arccos (\frac{2}{3}) + 2 πn

Divide both sides by

3

3 x = arccos (\frac{2}{3}) + 2 πn

Divide both sides by

3

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

Simplify

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

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

Solve

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

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

Divide both sides by

3

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

Divide both sides by

3

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

Simplify

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

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

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

cos (3 x) = - \frac{1}{2} : x = \frac{2 π}{9} + \frac{2 πn}{3}, x = \frac{4 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{2 π}{3} + 2 πn, 3 x = \frac{4 π}{3} + 2 πn

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

Solve

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

3 x = \frac{2 π}{3} + 2 πn

Divide both sides by

3

3 x = \frac{2 π}{3} + 2 πn

Divide both sides by

3

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

Simplify

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

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

\frac{\frac{2 π}{3}}{3} = \frac{2 π}{9}

\frac{\frac{2 π}{3}}{3}

Apply the fraction rule:

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

= \frac{2 π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{2 π}{9}

= \frac{2 π}{9} + \frac{2 πn}{3}

x = \frac{2 π}{9} + \frac{2 πn}{3}

x = \frac{2 π}{9} + \frac{2 πn}{3}

x = \frac{2 π}{9} + \frac{2 πn}{3}

Solve

3 x = \frac{4 π}{3} + 2 πn : x = \frac{4 π}{9} + \frac{2 πn}{3}

3 x = \frac{4 π}{3} + 2 πn

Divide both sides by

3

3 x = \frac{4 π}{3} + 2 πn

Divide both sides by

3

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

Simplify

\frac{3 x}{3} = \frac{\frac{4 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{4 π}{3}}{3} + \frac{2 πn}{3} : \frac{4 π}{9} + \frac{2 πn}{3}

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

\frac{\frac{4 π}{3}}{3} = \frac{4 π}{9}

\frac{\frac{4 π}{3}}{3}

Apply the fraction rule:

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

= \frac{4 π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{4 π}{9}

= \frac{4 π}{9} + \frac{2 πn}{3}

x = \frac{4 π}{9} + \frac{2 πn}{3}

x = \frac{4 π}{9} + \frac{2 πn}{3}

x = \frac{4 π}{9} + \frac{2 πn}{3}

x = \frac{2 π}{9} + \frac{2 πn}{3}, x = \frac{4 π}{9} + \frac{2 πn}{3}

Combine all the solutions

x = \frac{arccos ( \frac{2}{3} )}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{arccos ( \frac{2}{3} )}{3} + \frac{2 πn}{3}, x = \frac{2 π}{9} + \frac{2 πn}{3}, x = \frac{4 π}{9} + \frac{2 πn}{3}

Show solutions in decimal form

x = \frac{0.84106 \dots}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{0.84106 \dots}{3} + \frac{2 πn}{3}, x = \frac{2 π}{9} + \frac{2 πn}{3}, x = \frac{4 π}{9} + \frac{2 πn}{3}

Graph

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