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Popular Trigonometry >

6tan^2(x)-2cos^2(x)=cos(2x)

Solution

6 tan^{2} (x) - 2 cos^{2} (x) = cos (2 x)

Solution

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n

Solution steps

6 tan^{2} (x) - 2 cos^{2} (x) = cos (2 x)

Subtract

cos (2 x)

from both sides

6 tan^{2} (x) - 2 cos^{2} (x) - cos (2 x) = 0

Express with sin, cos

- cos (2 x) - 2 cos^{2} (x) + 6 tan^{2} (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

6 (\frac{sin ( x )}{cos ( x )})^{2} = \frac{6 sin ^{2} ( x )}{cos ^{2} ( x )}

6 (\frac{sin ( x )}{cos ( x )})^{2}

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

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

Apply exponent rule:

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

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

= 6 \cdot \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

Multiply fractions:

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

= \frac{sin ^{2} ( x ) \cdot 6}{cos ^{2} ( x )}

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

Convert element to fraction:

cos (2 x) = \frac{cos ( 2 x ) cos ^{2} ( x )}{cos ^{2} ( x )}, 2 cos^{2} (x) = \frac{2 cos ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

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

Since the denominators are equal, combine the fractions:

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

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

- cos (2 x) cos^{2} (x) - 2 cos^{2} (x) cos^{2} (x) + sin^{2} (x) \cdot 6 = - cos^{2} (x) cos (2 x) - 2 cos^{4} (x) + 6 sin^{2} (x)

- cos (2 x) cos^{2} (x) - 2 cos^{2} (x) cos^{2} (x) + sin^{2} (x) \cdot 6

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

2 cos^{2} (x) cos^{2} (x)

Apply exponent rule:

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

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

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

Add the numbers:

2 + 2 = 4

= 2 cos^{4} (x)

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

= \frac{- cos ^{2} ( x ) cos ( 2 x ) - 2 cos ^{4} ( x ) + 6 sin ^{2} ( x )}{cos ^{2} ( x )}

= \frac{- cos ^{2} ( x ) cos ( 2 x ) - 2 cos ^{4} ( x ) + 6 sin ^{2} ( x )}{cos ^{2} ( x )}

\frac{- 2 cos ^{4} ( x ) + 6 sin ^{2} ( x ) - cos ( 2 x ) cos ^{2} ( x )}{cos ^{2} ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- 2 cos^{4} (x) + 6 sin^{2} (x) - cos (2 x) cos^{2} (x) = 0

Rewrite using trig identities

- 2 cos^{4} (x) + 6 sin^{2} (x) - cos (2 x) cos^{2} (x)

Use the Double Angle identity:

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

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

Simplify

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

- 2 cos^{4} (x) + 6 sin^{2} (x) - cos^{2} (x) (cos^{2} (x) - sin^{2} (x))

Expand

- cos^{2} (x) (cos^{2} (x) - sin^{2} (x)) : - cos^{4} (x) + cos^{2} (x) sin^{2} (x)

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

Apply minus-plus rules

- (- a) = a

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

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

cos^{2} (x) cos^{2} (x)

Apply exponent rule:

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

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

= cos^{2 + 2} (x)

Add the numbers:

2 + 2 = 4

= cos^{4} (x)

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

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

Add similar elements:

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

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

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

Use the Pythagorean identity:

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

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

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

Simplify

- 3 cos^{4} (x) + 6 (1 - cos^{2} (x)) + cos^{2} (x) (1 - cos^{2} (x)) : - 4 cos^{4} (x) - 5 cos^{2} (x) + 6

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

Expand

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

6 (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 6, b = 1, c = cos^{2} (x)

= 6 \cdot 1 - 6 cos^{2} (x)

Multiply the numbers:

6 \cdot 1 = 6

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

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

Simplify

1 \cdot cos^{2} (x) - cos^{2} (x) cos^{2} (x) : cos^{2} (x) - cos^{4} (x)

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

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

1 \cdot cos^{2} (x)

Multiply:

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

= cos^{2} (x)

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

cos^{2} (x) cos^{2} (x)

Apply exponent rule:

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

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

= cos^{2 + 2} (x)

Add the numbers:

2 + 2 = 4

= cos^{4} (x)

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

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

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

Simplify

- 3 cos^{4} (x) + 6 - 6 cos^{2} (x) + cos^{2} (x) - cos^{4} (x) : - 4 cos^{4} (x) - 5 cos^{2} (x) + 6

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

Group like terms

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

Add similar elements:

- 6 cos^{2} (x) + cos^{2} (x) = - 5 cos^{2} (x)

= - 3 cos^{4} (x) - 5 cos^{2} (x) - cos^{4} (x) + 6

Add similar elements:

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

= - 4 cos^{4} (x) - 5 cos^{2} (x) + 6

= - 4 cos^{4} (x) - 5 cos^{2} (x) + 6

= - 4 cos^{4} (x) - 5 cos^{2} (x) + 6

6 - 4 cos^{4} (x) - 5 cos^{2} (x) = 0

Solve by substitution

6 - 4 cos^{4} (x) - 5 cos^{2} (x) = 0

Let:

cos (x) = u

6 - 4 u^{4} - 5 u^{2} = 0

6 - 4 u^{4} - 5 u^{2} = 0 : u = 2 i, u = - 2 i, u = \frac{3}{2}, u = - \frac{3}{2}

6 - 4 u^{4} - 5 u^{2} = 0

Write in the standard form

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

- 4 u^{4} - 5 u^{2} + 6 = 0

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

- 4 v^{2} - 5 v + 6 = 0

Solve

- 4 v^{2} - 5 v + 6 = 0 : v = - 2, v = \frac{3}{4}

- 4 v^{2} - 5 v + 6 = 0

Solve with the quadratic formula

- 4 v^{2} - 5 v + 6 = 0

Quadratic Equation Formula:

For

a = - 4, b = - 5, c = 6

v_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 4 ) \cdot 6}{2 ( - 4 )}

v_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 4 ) \cdot 6}{2 ( - 4 )}

(- 5)^{2} - 4 (- 4) \cdot 6 = 11

(- 5)^{2} - 4 (- 4) \cdot 6

Apply rule

- (- a) = a

= (- 5)^{2} + 4 \cdot 4 \cdot 6

Apply exponent rule:

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

n

is even

(- 5)^{2} = 5^{2}

= 5^{2} + 4 \cdot 4 \cdot 6

Multiply the numbers:

4 \cdot 4 \cdot 6 = 96

= 5^{2} + 96

5^{2} = 25

= 25 + 96

Add the numbers:

25 + 96 = 121

= 121

Factor the number:

121 = 1 1^{2}

= 1 1^{2}

Apply radical rule:

1 1^{2} = 11

= 11

v_{1, 2} = \frac{- ( - 5 ) \pm 11}{2 ( - 4 )}

Separate the solutions

v_{1} = \frac{- ( - 5 ) + 11}{2 ( - 4 )}, v_{2} = \frac{- ( - 5 ) - 11}{2 ( - 4 )}

v = \frac{- ( - 5 ) + 11}{2 ( - 4 )} : - 2

\frac{- ( - 5 ) + 11}{2 ( - 4 )}

Remove parentheses:

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

= \frac{5 + 11}{- 2 \cdot 4}

Add the numbers:

5 + 11 = 16

= \frac{16}{- 2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{16}{- 8}

Apply the fraction rule:

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

= - \frac{16}{8}

Divide the numbers:

\frac{16}{8} = 2

= - 2

v = \frac{- ( - 5 ) - 11}{2 ( - 4 )} : \frac{3}{4}

\frac{- ( - 5 ) - 11}{2 ( - 4 )}

Remove parentheses:

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

= \frac{5 - 11}{- 2 \cdot 4}

Subtract the numbers:

5 - 11 = - 6

= \frac{- 6}{- 2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 6}{- 8}

Apply the fraction rule:

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

= \frac{6}{8}

Cancel the common factor:

2

= \frac{3}{4}

The solutions to the quadratic equation are:

v = - 2, v = \frac{3}{4}

v = - 2, v = \frac{3}{4}

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = - 2 : u = 2 i, u = - 2 i

u^{2} = - 2

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = - 2, u = - - 2

Simplify

- 2 : 2 i

- 2

Apply radical rule:

- a = - 1 a

- 2 = - 1 2

= - 1 2

Apply imaginary number rule:

- 1 = i

= 2 i

Simplify

- - 2 : - 2 i

- - 2

Simplify

- 2 : 2 i

- 2

Apply radical rule:

- a = - 1 a

- 2 = - 1 2

= - 1 2

Apply imaginary number rule:

- 1 = i

= 2 i

= - 2 i

u = 2 i, u = - 2 i

Solve

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

u^{2} = \frac{3}{4}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

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

\frac{3}{4} = \frac{3}{2}

\frac{3}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{3}{2}

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

- \frac{3}{4}

Simplify

\frac{3}{4} : \frac{3}{2}

\frac{3}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{3}{2}

= - \frac{3}{2}

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

The solutions are

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

Substitute back

u = cos (x)

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

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

cos (x) = 2 i :

No Solution

cos (x) = 2 i

No Solution

cos (x) = - 2 i :

No Solution

cos (x) = - 2 i

No Solution

cos (x) = \frac{3}{2} : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

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

General solutions for

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

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

cos (x) = - \frac{3}{2} : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

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

General solutions for

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

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Graph

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