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Popular Trigonometry >

20cos^6(x)-57cos^4(x)+27cos^2(x)=0

Solution

20 cos^{6} (x) - 57 cos^{4} (x) + 27 cos^{2} (x) = 0

Solution

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 0.68471 \dots + 2 πn, x = 2 π - 0.68471 \dots + 2 πn, x = 2.45687 \dots + 2 πn, x = - 2.45687 \dots + 2 πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 39.23152 \dots^{\circ} + 36 0^{\circ} n, x = 320.76847 \dots^{\circ} + 36 0^{\circ} n, x = 140.76847 \dots^{\circ} + 36 0^{\circ} n, x = - 140.76847 \dots^{\circ} + 36 0^{\circ} n

Solution steps

20 cos^{6} (x) - 57 cos^{4} (x) + 27 cos^{2} (x) = 0

Solve by substitution

20 cos^{6} (x) - 57 cos^{4} (x) + 27 cos^{2} (x) = 0

Let:

cos (x) = u

20 u^{6} - 57 u^{4} + 27 u^{2} = 0

20 u^{6} - 57 u^{4} + 27 u^{2} = 0 : u = 0, u = \frac{3}{5}, u = - \frac{3}{5}, u = \frac{3}{2}, u = - \frac{3}{2}

20 u^{6} - 57 u^{4} + 27 u^{2} = 0

Rewrite the equation with

v = u^{2}, v^{2} = u^{4}

and

v^{3} = u^{6}

20 v^{3} - 57 v^{2} + 27 v = 0

Solve

20 v^{3} - 57 v^{2} + 27 v = 0 : v = 0, v = \frac{3}{5}, v = \frac{9}{4}

20 v^{3} - 57 v^{2} + 27 v = 0

Factor

20 v^{3} - 57 v^{2} + 27 v : v (5 v - 3) (4 v - 9)

20 v^{3} - 57 v^{2} + 27 v

Factor out common term

v : v (20 v^{2} - 57 v + 27)

20 v^{3} - 57 v^{2} + 27 v

Apply exponent rule:

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

v^{2} = vv

= 20 v^{2} v - 57 vv + 27 v

Factor out common term

v

= v (20 v^{2} - 57 v + 27)

= v (20 v^{2} - 57 v + 27)

Factor

20 v^{2} - 57 v + 27 : (5 v - 3) (4 v - 9)

20 v^{2} - 57 v + 27

Break the expression into groups

20 v^{2} - 57 v + 27

Definition

Factors of

540 : 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, 270, 540

540

Divisors (Factors)

Find the Prime factors of

540 : 2, 2, 3, 3, 3, 5

540

540

divides by

2540 = 270 \cdot 2

= 2 \cdot 270

270

divides by

2270 = 135 \cdot 2

= 2 \cdot 2 \cdot 135

135

divides by

3135 = 45 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 45

45

divides by

345 = 15 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 15

15

divides by

315 = 5 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 5

2, 3, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 5

Multiply the prime factors of

540 : 4, 6, 12, 9, 18, 36, 27, 54, 108, 10, 20, 15, 30, 60, 45, 90, 180, 135, 270

2 \cdot 2 = 4

2 \cdot 3 = 6

4, 6, 12, 9, 18, 36, 27, 54, 108, 10, 20, 15, 30, 60, 45, 90, 180, 135, 270

4, 6, 12, 9, 18, 36, 27, 54, 108, 10, 20, 15, 30, 60, 45, 90, 180, 135, 270

Add the prime factors:

2, 3, 5

Add 1 and the number

540

itself

1, 540

The factors of

540

1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, 270, 540

Negative factors of

540 : - 1, - 2, - 3, - 4, - 5, - 6, - 9, - 10, - 12, - 15, - 18, - 20, - 27, - 30, - 36, - 45, - 54, - 60, - 90, - 108, - 135, - 180, - 270, - 540

Multiply the factors by

- 1

to get the negative factors

- 1, - 2, - 3, - 4, - 5, - 6, - 9, - 10, - 12, - 15, - 18, - 20, - 27, - 30, - 36, - 45, - 54, - 60, - 90, - 108, - 135, - 180, - 270, - 540

For every two factors such that

u * v = 540,

check if

u + v = - 57

Check

u = 1, v = 540 : u * v = 540, u + v = 541 \Rightarrow

FalseCheck

u = 2, v = 270 : u * v = 540, u + v = 272 \Rightarrow

False

u = - 12, v = - 45

Group into

(a x^{2} + ux) + (vx + c)

(20 v^{2} - 12 v) + (- 45 v + 27)

= (20 v^{2} - 12 v) + (- 45 v + 27)

Factor out

4 v

from

20 v^{2} - 12 v : 4 v (5 v - 3)

20 v^{2} - 12 v

Apply exponent rule:

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

v^{2} = vv

= 20 vv - 12 v

Rewrite

12

4 \cdot 3

Rewrite

20

4 \cdot 5

= 4 \cdot 5 vv - 4 \cdot 3 v

Factor out common term

4 v

= 4 v (5 v - 3)

Factor out

- 9

from

- 45 v + 27 : - 9 (5 v - 3)

- 45 v + 27

Rewrite

27

9 \cdot 3

Rewrite

45

9 \cdot 5

= - 9 \cdot 5 v + 9 \cdot 3

Factor out common term

- 9

= - 9 (5 v - 3)

= 4 v (5 v - 3) - 9 (5 v - 3)

Factor out common term

5 v - 3

= (5 v - 3) (4 v - 9)

= v (5 v - 3) (4 v - 9)

v (5 v - 3) (4 v - 9) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

v = 0 or 5 v - 3 = 0 or 4 v - 9 = 0

Solve

5 v - 3 = 0 : v = \frac{3}{5}

5 v - 3 = 0

Move

3

to the right side

5 v - 3 = 0

Add

3

to both sides

5 v - 3 + 3 = 0 + 3

Simplify

5 v = 3

5 v = 3

Divide both sides by

5

5 v = 3

Divide both sides by

5

\frac{5 v}{5} = \frac{3}{5}

Simplify

v = \frac{3}{5}

v = \frac{3}{5}

Solve

4 v - 9 = 0 : v = \frac{9}{4}

4 v - 9 = 0

Move

9

to the right side

4 v - 9 = 0

Add

9

to both sides

4 v - 9 + 9 = 0 + 9

Simplify

4 v = 9

4 v = 9

Divide both sides by

4

4 v = 9

Divide both sides by

4

\frac{4 v}{4} = \frac{9}{4}

Simplify

v = \frac{9}{4}

v = \frac{9}{4}

The solutions are

v = 0, v = \frac{3}{5}, v = \frac{9}{4}

v = 0, v = \frac{3}{5}, v = \frac{9}{4}

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = 0 : u = 0

u^{2} = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

u = 0

Solve

u^{2} = \frac{3}{5} : u = \frac{3}{5}, u = - \frac{3}{5}

u^{2} = \frac{3}{5}

For

x^{2} = f (a)

the solutions are

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

u = \frac{3}{5}, u = - \frac{3}{5}

Solve

u^{2} = \frac{9}{4} : u = \frac{3}{2}, u = - \frac{3}{2}

u^{2} = \frac{9}{4}

For

x^{2} = f (a)

the solutions are

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

u = \frac{9}{4}, u = - \frac{9}{4}

\frac{9}{4} = \frac{3}{2}

\frac{9}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{9}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{9}{2}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

= \frac{3}{2}

- \frac{9}{4} = - \frac{3}{2}

- \frac{9}{4}

Simplify

\frac{9}{4} : \frac{3}{2}

\frac{9}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{9}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{9}{2}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

= \frac{3}{2}

= - \frac{3}{2}

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

The solutions are

u = 0, u = \frac{3}{5}, u = - \frac{3}{5}, u = \frac{3}{2}, u = - \frac{3}{2}

Substitute back

u = cos (x)

cos (x) = 0, cos (x) = \frac{3}{5}, cos (x) = - \frac{3}{5}, cos (x) = \frac{3}{2}, cos (x) = - \frac{3}{2}

cos (x) = 0, cos (x) = \frac{3}{5}, cos (x) = - \frac{3}{5}, cos (x) = \frac{3}{2}, cos (x) = - \frac{3}{2}

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

cos (x) = 0

General solutions for

cos (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}

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

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

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

cos (x) = \frac{3}{5}

Apply trig inverse properties

cos (x) = \frac{3}{5}

General solutions for

cos (x) = \frac{3}{5}

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

x = arccos (\frac{3}{5}) + 2 πn, x = 2 π - arccos (\frac{3}{5}) + 2 πn

x = arccos (\frac{3}{5}) + 2 πn, x = 2 π - arccos (\frac{3}{5}) + 2 πn

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

cos (x) = - \frac{3}{5}

Apply trig inverse properties

cos (x) = - \frac{3}{5}

General solutions for

cos (x) = - \frac{3}{5}

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

x = arccos (- \frac{3}{5}) + 2 πn, x = - arccos (- \frac{3}{5}) + 2 πn

x = arccos (- \frac{3}{5}) + 2 πn, x = - arccos (- \frac{3}{5}) + 2 πn

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

No Solution

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

- 1 \leq cos (x) \leq 1

No Solution

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

No Solution

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

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = arccos (\frac{3}{5}) + 2 πn, x = 2 π - arccos (\frac{3}{5}) + 2 πn, x = arccos (- \frac{3}{5}) + 2 πn, x = - arccos (- \frac{3}{5}) + 2 πn

Show solutions in decimal form

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 0.68471 \dots + 2 πn, x = 2 π - 0.68471 \dots + 2 πn, x = 2.45687 \dots + 2 πn, x = - 2.45687 \dots + 2 πn

Graph

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