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Popular Trigonometry >

5sin(2x)=9tan(x)

Solution

5 sin (2 x) = 9 tan (x)

Solution

x = 2 πn, x = π + 2 πn, x = 2.81984 \dots + 2 πn, x = - 2.81984 \dots + 2 πn, x = 0.32175 \dots + 2 πn, x = 2 π - 0.32175 \dots + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 161.56505 \dots^{\circ} + 36 0^{\circ} n, x = - 161.56505 \dots^{\circ} + 36 0^{\circ} n, x = 18.43494 \dots^{\circ} + 36 0^{\circ} n, x = 341.56505 \dots^{\circ} + 36 0^{\circ} n

Solution steps

5 sin (2 x) = 9 tan (x)

Subtract

9 tan (x)

from both sides

5 sin (2 x) - 9 tan (x) = 0

Express with sin, cos

5 sin (2 x) - 9 tan (x)

Use the basic trigonometric identity:

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

= 5 sin (2 x) - 9 \cdot \frac{sin ( x )}{cos ( x )}

Simplify

5 sin (2 x) - 9 \cdot \frac{sin ( x )}{cos ( x )} : \frac{5 sin ( 2 x ) cos ( x ) - 9 sin ( x )}{cos ( x )}

5 sin (2 x) - 9 \cdot \frac{sin ( x )}{cos ( x )}

Multiply

9 \cdot \frac{sin ( x )}{cos ( x )} : \frac{9 sin ( x )}{cos ( x )}

9 \cdot \frac{sin ( x )}{cos ( x )}

Multiply fractions:

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

= \frac{sin ( x ) \cdot 9}{cos ( x )}

= 5 sin (2 x) - \frac{9 sin ( x )}{cos ( x )}

Convert element to fraction:

5 sin (2 x) = \frac{5 sin ( 2 x ) cos ( x )}{cos ( x )}

= \frac{5 sin ( 2 x ) cos ( x )}{cos ( x )} - \frac{sin ( x ) \cdot 9}{cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{5 sin ( 2 x ) cos ( x ) - sin ( x ) \cdot 9}{cos ( x )}

= \frac{5 sin ( 2 x ) cos ( x ) - 9 sin ( x )}{cos ( x )}

\frac{- 9 sin ( x ) + 5 cos ( x ) sin ( 2 x )}{cos ( x )} = 0

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

- 9 sin (x) + 5 cos (x) sin (2 x) = 0

Rewrite using trig identities

- 9 sin (x) + 5 cos (x) sin (2 x)

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= - 9 sin (x) + 5 cos (x) \cdot 2 sin (x) cos (x)

5 cos (x) \cdot 2 sin (x) cos (x) = 10 cos^{2} (x) sin (x)

5 cos (x) \cdot 2 sin (x) cos (x)

Multiply the numbers:

5 \cdot 2 = 10

= 10 cos (x) sin (x) cos (x)

Apply exponent rule:

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

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

= 10 sin (x) cos^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= 10 sin (x) cos^{2} (x)

= - 9 sin (x) + 10 cos^{2} (x) sin (x)

- 9 sin (x) + 10 cos^{2} (x) sin (x) = 0

Factor

- 9 sin (x) + 10 cos^{2} (x) sin (x) : sin (x) (10 cos (x) + 3) (10 cos (x) - 3)

- 9 sin (x) + 10 cos^{2} (x) sin (x)

Factor out common term

sin (x)

= sin (x) (- 9 + 10 cos^{2} (x))

Factor

10 cos^{2} (x) - 9 : (10 cos (x) + 3) (10 cos (x) - 3)

10 cos^{2} (x) - 9

Rewrite

10 cos^{2} (x) - 9

(10 cos (x))^{2} - 3^{2}

10 cos^{2} (x) - 9

Apply radical rule:

a = (a)^{2}

10 = (10)^{2}

= (10)^{2} cos^{2} (x) - 9

Rewrite

9

3^{2}

= (10)^{2} cos^{2} (x) - 3^{2}

Apply exponent rule:

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

(10)^{2} cos^{2} (x) = (10 cos (x))^{2}

= (10 cos (x))^{2} - 3^{2}

= (10 cos (x))^{2} - 3^{2}

Apply Difference of Two Squares Formula:

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

(10 cos (x))^{2} - 3^{2} = (10 cos (x) + 3) (10 cos (x) - 3)

= (10 cos (x) + 3) (10 cos (x) - 3)

= sin (x) (10 cos (x) + 3) (10 cos (x) - 3)

sin (x) (10 cos (x) + 3) (10 cos (x) - 3) = 0

Solving each part separately

sin (x) = 0 or 10 cos (x) + 3 = 0 or 10 cos (x) - 3 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

10 cos (x) + 3 = 0 : x = arccos (- \frac{3 10}{10}) + 2 πn, x = - arccos (- \frac{3 10}{10}) + 2 πn

10 cos (x) + 3 = 0

Move

3

to the right side

10 cos (x) + 3 = 0

Subtract

3

from both sides

10 cos (x) + 3 - 3 = 0 - 3

Simplify

10 cos (x) = - 3

10 cos (x) = - 3

Divide both sides by

10

10 cos (x) = - 3

Divide both sides by

10

\frac{10 cos ( x )}{10} = \frac{- 3}{10}

Simplify

\frac{10 cos ( x )}{10} = \frac{- 3}{10}

Simplify

\frac{10 cos ( x )}{10} : cos (x)

\frac{10 cos ( x )}{10}

Cancel the common factor:

10

= cos (x)

Simplify

\frac{- 3}{10} : - \frac{3 10}{10}

\frac{- 3}{10}

Apply the fraction rule:

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

= - \frac{3}{10}

Rationalize

- \frac{3}{10} : - \frac{3 10}{10}

- \frac{3}{10}

Multiply by the conjugate

\frac{10}{10}

= - \frac{3 10}{10 10}

1010 = 10

1010

Apply radical rule:

a a = a

1010 = 10

= 10

= - \frac{3 10}{10}

= - \frac{3 10}{10}

cos (x) = - \frac{3 10}{10}

cos (x) = - \frac{3 10}{10}

cos (x) = - \frac{3 10}{10}

Apply trig inverse properties

cos (x) = - \frac{3 10}{10}

General solutions for

cos (x) = - \frac{3 10}{10}

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

x = arccos (- \frac{3 10}{10}) + 2 πn, x = - arccos (- \frac{3 10}{10}) + 2 πn

x = arccos (- \frac{3 10}{10}) + 2 πn, x = - arccos (- \frac{3 10}{10}) + 2 πn

10 cos (x) - 3 = 0 : x = arccos (\frac{3 10}{10}) + 2 πn, x = 2 π - arccos (\frac{3 10}{10}) + 2 πn

10 cos (x) - 3 = 0

Move

3

to the right side

10 cos (x) - 3 = 0

Add

3

to both sides

10 cos (x) - 3 + 3 = 0 + 3

Simplify

10 cos (x) = 3

10 cos (x) = 3

Divide both sides by

10

10 cos (x) = 3

Divide both sides by

10

\frac{10 cos ( x )}{10} = \frac{3}{10}

Simplify

\frac{10 cos ( x )}{10} = \frac{3}{10}

Simplify

\frac{10 cos ( x )}{10} : cos (x)

\frac{10 cos ( x )}{10}

Cancel the common factor:

10

= cos (x)

Simplify

\frac{3}{10} : \frac{3 10}{10}

\frac{3}{10}

Multiply by the conjugate

\frac{10}{10}

= \frac{3 10}{10 10}

1010 = 10

1010

Apply radical rule:

a a = a

1010 = 10

= 10

= \frac{3 10}{10}

cos (x) = \frac{3 10}{10}

cos (x) = \frac{3 10}{10}

cos (x) = \frac{3 10}{10}

Apply trig inverse properties

cos (x) = \frac{3 10}{10}

General solutions for

cos (x) = \frac{3 10}{10}

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

x = arccos (\frac{3 10}{10}) + 2 πn, x = 2 π - arccos (\frac{3 10}{10}) + 2 πn

x = arccos (\frac{3 10}{10}) + 2 πn, x = 2 π - arccos (\frac{3 10}{10}) + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arccos (- \frac{3 10}{10}) + 2 πn, x = - arccos (- \frac{3 10}{10}) + 2 πn, x = arccos (\frac{3 10}{10}) + 2 πn, x = 2 π - arccos (\frac{3 10}{10}) + 2 πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 2.81984 \dots + 2 πn, x = - 2.81984 \dots + 2 πn, x = 0.32175 \dots + 2 πn, x = 2 π - 0.32175 \dots + 2 πn

Graph

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