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Popular >

(2sin(x)+5)(3cos(x)+3)(tan^2(x)-2)=0

Solution

(2 sin (x) + 5) (3 cos (x) + 3) (tan^{2} (x) - 2) = 0

Solution

x = π + 2 πn, x = 0.95531 \dots + πn, x = - 0.95531 \dots + πn

+1

Degrees

x = 18 0^{\circ} + 36 0^{\circ} n, x = 54.73561 \dots^{\circ} + 18 0^{\circ} n, x = - 54.73561 \dots^{\circ} + 18 0^{\circ} n

Solution steps

(2 sin (x) + 5) (3 cos (x) + 3) (tan^{2} (x) - 2) = 0

Solving each part separately

2 sin (x) + 5 = 0 or 3 cos (x) + 3 = 0 or tan^{2} (x) - 2 = 0

2 sin (x) + 5 = 0 :

No Solution

2 sin (x) + 5 = 0

Move

5

to the right side

2 sin (x) + 5 = 0

Subtract

5

from both sides

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

Simplify

2 sin (x) = - 5

2 sin (x) = - 5

Divide both sides by

2

2 sin (x) = - 5

Divide both sides by

2

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

Simplify

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

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

- 1 \leq sin (x) \leq 1

No Solution

3 cos (x) + 3 = 0 : x = π + 2 πn

3 cos (x) + 3 = 0

Move

3

to the right side

3 cos (x) + 3 = 0

Subtract

3

from both sides

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

Simplify

3 cos (x) = - 3

3 cos (x) = - 3

Divide both sides by

3

3 cos (x) = - 3

Divide both sides by

3

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

Simplify

cos (x) = - 1

cos (x) = - 1

General solutions for

cos (x) = - 1

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}

x = π + 2 πn

x = π + 2 πn

tan^{2} (x) - 2 = 0 : x = arctan (2) + πn, x = arctan (- 2) + πn

tan^{2} (x) - 2 = 0

Solve by substitution

tan^{2} (x) - 2 = 0

Let:

tan (x) = u

u^{2} - 2 = 0

u^{2} - 2 = 0 : u = 2, u = - 2

u^{2} - 2 = 0

Move

2

to the right side

u^{2} - 2 = 0

Add

2

to both sides

u^{2} - 2 + 2 = 0 + 2

Simplify

u^{2} = 2

u^{2} = 2

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = 2, u = - 2

Substitute back

u = tan (x)

tan (x) = 2, tan (x) = - 2

tan (x) = 2, tan (x) = - 2

tan (x) = 2 : x = arctan (2) + πn

tan (x) = 2

Apply trig inverse properties

tan (x) = 2

General solutions for

tan (x) = 2

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (2) + πn

x = arctan (2) + πn

tan (x) = - 2 : x = arctan (- 2) + πn

tan (x) = - 2

Apply trig inverse properties

tan (x) = - 2

General solutions for

tan (x) = - 2

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- 2) + πn

x = arctan (- 2) + πn

Combine all the solutions

x = arctan (2) + πn, x = arctan (- 2) + πn

Combine all the solutions

x = π + 2 πn, x = arctan (2) + πn, x = arctan (- 2) + πn

Show solutions in decimal form

x = π + 2 πn, x = 0.95531 \dots + πn, x = - 0.95531 \dots + πn

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