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Popular Trigonometry >

5sin^2(x)cos(7x)-cos(7x)=0

Solution

5 sin^{2} (x) cos (7 x) - cos (7 x) = 0

Solution

x = \frac{π}{14} + \frac{2 πn}{7}, x = \frac{3 π}{14} + \frac{2 πn}{7}, x = - 0.46364 \dots + 2 πn, x = π + 0.46364 \dots + 2 πn, x = 0.46364 \dots + 2 πn, x = π - 0.46364 \dots + 2 πn

Degrees

x = 12.85714 \dots^{\circ} + 51.42857 \dots^{\circ} n, x = 38.57142 \dots^{\circ} + 51.42857 \dots^{\circ} n, x = - 26.56505 \dots^{\circ} + 36 0^{\circ} n, x = 206.56505 \dots^{\circ} + 36 0^{\circ} n, x = 26.56505 \dots^{\circ} + 36 0^{\circ} n, x = 153.43494 \dots^{\circ} + 36 0^{\circ} n

Solution steps

5 sin^{2} (x) cos (7 x) - cos (7 x) = 0

Factor

5 sin^{2} (x) cos (7 x) - cos (7 x) : cos (7 x) (5 sin (x) + 1) (5 sin (x) - 1)

5 sin^{2} (x) cos (7 x) - cos (7 x)

Factor out common term

cos (7 x)

= cos (7 x) (5 sin^{2} (x) - 1)

Factor

5 sin^{2} (x) - 1 : (5 sin (x) + 1) (5 sin (x) - 1)

5 sin^{2} (x) - 1

Rewrite

5 sin^{2} (x) - 1

(5 sin (x))^{2} - 1^{2}

5 sin^{2} (x) - 1

Apply radical rule:

a = (a)^{2}

5 = (5)^{2}

= (5)^{2} sin^{2} (x) - 1

Rewrite

1

1^{2}

= (5)^{2} sin^{2} (x) - 1^{2}

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

(5)^{2} sin^{2} (x) = (5 sin (x))^{2}

= (5 sin (x))^{2} - 1^{2}

= (5 sin (x))^{2} - 1^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

(5 sin (x))^{2} - 1^{2} = (5 sin (x) + 1) (5 sin (x) - 1)

= (5 sin (x) + 1) (5 sin (x) - 1)

= cos (7 x) (5 sin (x) + 1) (5 sin (x) - 1)

cos (7 x) (5 sin (x) + 1) (5 sin (x) - 1) = 0

Solving each part separately

cos (7 x) = 0 or 5 sin (x) + 1 = 0 or 5 sin (x) - 1 = 0

cos (7 x) = 0 : x = \frac{π}{14} + \frac{2 πn}{7}, x = \frac{3 π}{14} + \frac{2 πn}{7}

cos (7 x) = 0

General solutions for

cos (7 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}

7 x = \frac{π}{2} + 2 πn, 7 x = \frac{3 π}{2} + 2 πn

7 x = \frac{π}{2} + 2 πn, 7 x = \frac{3 π}{2} + 2 πn

Solve

7 x = \frac{π}{2} + 2 πn : x = \frac{π}{14} + \frac{2 πn}{7}

7 x = \frac{π}{2} + 2 πn

Divide both sides by

7

7 x = \frac{π}{2} + 2 πn

Divide both sides by

7

\frac{7 x}{7} = \frac{\frac{π}{2}}{7} + \frac{2 πn}{7}

Simplify

\frac{7 x}{7} = \frac{\frac{π}{2}}{7} + \frac{2 πn}{7}

Simplify

\frac{7 x}{7} : x

\frac{7 x}{7}

Divide the numbers:

\frac{7}{7} = 1

= x

Simplify

\frac{\frac{π}{2}}{7} + \frac{2 πn}{7} : \frac{π}{14} + \frac{2 πn}{7}

\frac{\frac{π}{2}}{7} + \frac{2 πn}{7}

\frac{\frac{π}{2}}{7} = \frac{π}{14}

\frac{\frac{π}{2}}{7}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 7}

Multiply the numbers:

2 \cdot 7 = 14

= \frac{π}{14}

= \frac{π}{14} + \frac{2 πn}{7}

x = \frac{π}{14} + \frac{2 πn}{7}

x = \frac{π}{14} + \frac{2 πn}{7}

x = \frac{π}{14} + \frac{2 πn}{7}

Solve

7 x = \frac{3 π}{2} + 2 πn : x = \frac{3 π}{14} + \frac{2 πn}{7}

7 x = \frac{3 π}{2} + 2 πn

Divide both sides by

7

7 x = \frac{3 π}{2} + 2 πn

Divide both sides by

7

\frac{7 x}{7} = \frac{\frac{3 π}{2}}{7} + \frac{2 πn}{7}

Simplify

\frac{7 x}{7} = \frac{\frac{3 π}{2}}{7} + \frac{2 πn}{7}

Simplify

\frac{7 x}{7} : x

\frac{7 x}{7}

Divide the numbers:

\frac{7}{7} = 1

= x

Simplify

\frac{\frac{3 π}{2}}{7} + \frac{2 πn}{7} : \frac{3 π}{14} + \frac{2 πn}{7}

\frac{\frac{3 π}{2}}{7} + \frac{2 πn}{7}

\frac{\frac{3 π}{2}}{7} = \frac{3 π}{14}

\frac{\frac{3 π}{2}}{7}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 7}

Multiply the numbers:

2 \cdot 7 = 14

= \frac{3 π}{14}

= \frac{3 π}{14} + \frac{2 πn}{7}

x = \frac{3 π}{14} + \frac{2 πn}{7}

x = \frac{3 π}{14} + \frac{2 πn}{7}

x = \frac{3 π}{14} + \frac{2 πn}{7}

x = \frac{π}{14} + \frac{2 πn}{7}, x = \frac{3 π}{14} + \frac{2 πn}{7}

5 sin (x) + 1 = 0 : x = arcsin (- \frac{5}{5}) + 2 πn, x = π + arcsin (\frac{5}{5}) + 2 πn

5 sin (x) + 1 = 0

Move

1

to the right side

5 sin (x) + 1 = 0

Subtract

1

from both sides

5 sin (x) + 1 - 1 = 0 - 1

Simplify

5 sin (x) = - 1

5 sin (x) = - 1

Divide both sides by

5

5 sin (x) = - 1

Divide both sides by

5

\frac{5 sin ( x )}{5} = \frac{- 1}{5}

Simplify

\frac{5 sin ( x )}{5} = \frac{- 1}{5}

Simplify

\frac{5 sin ( x )}{5} : sin (x)

\frac{5 sin ( x )}{5}

Cancel the common factor:

5

= sin (x)

Simplify

\frac{- 1}{5} : - \frac{5}{5}

\frac{- 1}{5}

Apply the fraction rule:

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

= - \frac{1}{5}

Rationalize

- \frac{1}{5} : - \frac{5}{5}

- \frac{1}{5}

Multiply by the conjugate

\frac{5}{5}

= - \frac{1 \cdot 5}{5 5}

1 \cdot 5 = 5

55 = 5

55

Apply radical rule:

a a = a

55 = 5

= 5

= - \frac{5}{5}

= - \frac{5}{5}

sin (x) = - \frac{5}{5}

sin (x) = - \frac{5}{5}

sin (x) = - \frac{5}{5}

Apply trig inverse properties

sin (x) = - \frac{5}{5}

General solutions for

sin (x) = - \frac{5}{5}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

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

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

5 sin (x) - 1 = 0 : x = arcsin (\frac{5}{5}) + 2 πn, x = π - arcsin (\frac{5}{5}) + 2 πn

5 sin (x) - 1 = 0

Move

1

to the right side

5 sin (x) - 1 = 0

Add

1

to both sides

5 sin (x) - 1 + 1 = 0 + 1

Simplify

5 sin (x) = 1

5 sin (x) = 1

Divide both sides by

5

5 sin (x) = 1

Divide both sides by

5

\frac{5 sin ( x )}{5} = \frac{1}{5}

Simplify

\frac{5 sin ( x )}{5} = \frac{1}{5}

Simplify

\frac{5 sin ( x )}{5} : sin (x)

\frac{5 sin ( x )}{5}

Cancel the common factor:

5

= sin (x)

Simplify

\frac{1}{5} : \frac{5}{5}

\frac{1}{5}

Multiply by the conjugate

\frac{5}{5}

= \frac{1 \cdot 5}{5 5}

1 \cdot 5 = 5

55 = 5

55

Apply radical rule:

a a = a

55 = 5

= 5

= \frac{5}{5}

sin (x) = \frac{5}{5}

sin (x) = \frac{5}{5}

sin (x) = \frac{5}{5}

Apply trig inverse properties

sin (x) = \frac{5}{5}

General solutions for

sin (x) = \frac{5}{5}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

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

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

Combine all the solutions

x = \frac{π}{14} + \frac{2 πn}{7}, x = \frac{3 π}{14} + \frac{2 πn}{7}, x = arcsin (- \frac{5}{5}) + 2 πn, x = π + arcsin (\frac{5}{5}) + 2 πn, x = arcsin (\frac{5}{5}) + 2 πn, x = π - arcsin (\frac{5}{5}) + 2 πn

Show solutions in decimal form

x = \frac{π}{14} + \frac{2 πn}{7}, x = \frac{3 π}{14} + \frac{2 πn}{7}, x = - 0.46364 \dots + 2 πn, x = π + 0.46364 \dots + 2 πn, x = 0.46364 \dots + 2 πn, x = π - 0.46364 \dots + 2 πn

Graph

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