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Popular >

9cos^2(x)tan(x)=4tan(x)

Solution

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

Solution

x = πn, x = 2.30052 \dots + 2 πn, x = - 2.30052 \dots + 2 πn, x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 131.81031 \dots^{\circ} + 36 0^{\circ} n, x = - 131.81031 \dots^{\circ} + 36 0^{\circ} n, x = 48.18968 \dots^{\circ} + 36 0^{\circ} n, x = 311.81031 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

4 tan (x)

from both sides

9 cos^{2} (x) tan (x) - 4 tan (x) = 0

Factor

9 cos^{2} (x) tan (x) - 4 tan (x) : tan (x) (3 cos (x) + 2) (3 cos (x) - 2)

9 cos^{2} (x) tan (x) - 4 tan (x)

Factor out common term

tan (x)

= tan (x) (9 cos^{2} (x) - 4)

Factor

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

9 cos^{2} (x) - 4

Rewrite

9 cos^{2} (x) - 4

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

9 cos^{2} (x) - 4

Rewrite

9

3^{2}

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

Rewrite

4

2^{2}

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

Apply exponent rule:

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

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

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

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

Apply Difference of Two Squares Formula:

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

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

= (3 cos (x) + 2) (3 cos (x) - 2)

= tan (x) (3 cos (x) + 2) (3 cos (x) - 2)

tan (x) (3 cos (x) + 2) (3 cos (x) - 2) = 0

Solving each part separately

tan (x) = 0 or 3 cos (x) + 2 = 0 or 3 cos (x) - 2 = 0

tan (x) = 0 : x = πn

tan (x) = 0

General solutions for

tan (x) = 0

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 0 + πn

x = 0 + πn

Solve

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

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

3 cos (x) + 2 = 0

Move

2

to the right side

3 cos (x) + 2 = 0

Subtract

2

from both sides

3 cos (x) + 2 - 2 = 0 - 2

Simplify

3 cos (x) = - 2

3 cos (x) = - 2

Divide both sides by

3

3 cos (x) = - 2

Divide both sides by

3

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

Simplify

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

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

Apply trig inverse properties

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

General solutions for

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

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

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

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

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

3 cos (x) - 2 = 0

Move

2

to the right side

3 cos (x) - 2 = 0

Add

2

to both sides

3 cos (x) - 2 + 2 = 0 + 2

Simplify

3 cos (x) = 2

3 cos (x) = 2

Divide both sides by

3

3 cos (x) = 2

Divide both sides by

3

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

Simplify

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

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

Apply trig inverse properties

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

General solutions for

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

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

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

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

Combine all the solutions

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

Show solutions in decimal form

x = πn, x = 2.30052 \dots + 2 πn, x = - 2.30052 \dots + 2 πn, x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

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