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Popular Trigonometry >

4tan(x)sin^2(x)+3=4sin^2(x)+3tan(x)

Solution

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

Solution

x = \frac{π}{4} + πn, x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

4 sin^{2} (x) + 3 tan (x)

from both sides

4 sin^{2} (x) tan (x) + 3 - 4 sin^{2} (x) - 3 tan (x) = 0

Factor

4 sin^{2} (x) tan (x) + 3 - 4 sin^{2} (x) - 3 tan (x) : (- 1 + tan (x)) (2 sin (x) + 3) (2 sin (x) - 3)

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

Rewrite as

= 4 sin^{2} (x) tan (x) + 3 - 1 \cdot 4 sin^{2} (x) - 3 tan (x)

Factor out common term

4 sin^{2} (x)

= 4 sin^{2} (x) (tan (x) - 1) + 3 - 3 tan (x)

Rewrite as

= 4 sin^{2} (x) (- 1 + tan (x)) + 3 \cdot 1 - 3 tan (x)

Factor out common term

3

= 4 sin^{2} (x) (- 1 + tan (x)) + 3 (1 - tan (x))

Rewrite as

= 4 (- 1 + tan (x)) sin^{2} (x) - 3 (- 1 + tan (x))

Factor out common term

(- 1 + tan (x))

= (- 1 + tan (x)) (4 sin^{2} (x) - 3)

Factor

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

4 sin^{2} (x) - 3

Rewrite

4 sin^{2} (x) - 3

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

4 sin^{2} (x) - 3

Rewrite

4

2^{2}

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

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

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

Apply exponent rule:

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

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

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

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

Apply Difference of Two Squares Formula:

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

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

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

= (tan (x) - 1) (2 sin (x) + 3) (2 sin (x) - 3)

(- 1 + tan (x)) (2 sin (x) + 3) (2 sin (x) - 3) = 0

Solving each part separately

- 1 + tan (x) = 0 or 2 sin (x) + 3 = 0 or 2 sin (x) - 3 = 0

- 1 + tan (x) = 0 : x = \frac{π}{4} + πn

- 1 + tan (x) = 0

Move

1

to the right side

- 1 + tan (x) = 0

Add

1

to both sides

- 1 + tan (x) + 1 = 0 + 1

Simplify

tan (x) = 1

tan (x) = 1

General solutions for

tan (x) = 1

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 = \frac{π}{4} + πn

x = \frac{π}{4} + πn

2 sin (x) + 3 = 0 : x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

2 sin (x) + 3 = 0

Move

3

to the right side

2 sin (x) + 3 = 0

Subtract

3

from both sides

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

Simplify

2 sin (x) = - 3

2 sin (x) = - 3

Divide both sides by

2

2 sin (x) = - 3

Divide both sides by

2

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

Simplify

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

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

General solutions for

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

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 = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

2 sin (x) - 3 = 0

Move

3

to the right side

2 sin (x) - 3 = 0

Add

3

to both sides

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

Simplify

2 sin (x) = 3

2 sin (x) = 3

Divide both sides by

2

2 sin (x) = 3

Divide both sides by

2

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

Simplify

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

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

General solutions for

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

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

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

Combine all the solutions

x = \frac{π}{4} + πn, x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

Graph

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