What is the general solution for 9sin^2(x)tan(x)=16tan(x) ?

The general solution for 9sin^2(x)tan(x)=16tan(x) is x=pin

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Popular >

9sin^2(x)tan(x)=16tan(x)

Solution

9 sin^{2} (x) tan (x) = 16 tan (x)

Solution

x = πn

Degrees

x = 0^{\circ} + 18 0^{\circ} n

Solution steps

9 sin^{2} (x) tan (x) = 16 tan (x)

Subtract

16 tan (x)

from both sides

9 sin^{2} (x) tan (x) - 16 tan (x) = 0

Factor

9 sin^{2} (x) tan (x) - 16 tan (x) : tan (x) (3 sin (x) + 4) (3 sin (x) - 4)

9 sin^{2} (x) tan (x) - 16 tan (x)

Factor out common term

tan (x)

= tan (x) (9 sin^{2} (x) - 16)

Factor

9 sin^{2} (x) - 16 : (3 sin (x) + 4) (3 sin (x) - 4)

9 sin^{2} (x) - 16

Rewrite

9 sin^{2} (x) - 16

(3 sin (x))^{2} - 4^{2}

9 sin^{2} (x) - 16

Rewrite

9

3^{2}

= 3^{2} sin^{2} (x) - 16

Rewrite

16

4^{2}

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

Apply exponent rule:

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

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

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

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

Apply Difference of Two Squares Formula:

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

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

= (3 sin (x) + 4) (3 sin (x) - 4)

= tan (x) (3 sin (x) + 4) (3 sin (x) - 4)

tan (x) (3 sin (x) + 4) (3 sin (x) - 4) = 0

Solving each part separately

tan (x) = 0 or 3 sin (x) + 4 = 0 or 3 sin (x) - 4 = 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 sin (x) + 4 = 0 :

No Solution

3 sin (x) + 4 = 0

Move

4

to the right side

3 sin (x) + 4 = 0

Subtract

4

from both sides

3 sin (x) + 4 - 4 = 0 - 4

Simplify

3 sin (x) = - 4

3 sin (x) = - 4

Divide both sides by

3

3 sin (x) = - 4

Divide both sides by

3

\frac{3 sin ( x )}{3} = \frac{- 4}{3}

Simplify

sin (x) = - \frac{4}{3}

sin (x) = - \frac{4}{3}

- 1 \leq sin (x) \leq 1

No Solution

3 sin (x) - 4 = 0 :

No Solution

3 sin (x) - 4 = 0

Move

4

to the right side

3 sin (x) - 4 = 0

Add

4

to both sides

3 sin (x) - 4 + 4 = 0 + 4

Simplify

3 sin (x) = 4

3 sin (x) = 4

Divide both sides by

3

3 sin (x) = 4

Divide both sides by

3

\frac{3 sin ( x )}{3} = \frac{4}{3}

Simplify

sin (x) = \frac{4}{3}

sin (x) = \frac{4}{3}

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = πn

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Frequently Asked Questions (FAQ)

What is the general solution for 9sin^2(x)tan(x)=16tan(x) ?
The general solution for 9sin^2(x)tan(x)=16tan(x) is x=pin