What is the general solution for 3(1+sin(θ))=3(1-sin(θ)) ?

The general solution for 3(1+sin(θ))=3(1-sin(θ)) is θ=2pin,θ=pi+2pin

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Popular >

3(1+sin(θ))=3(1-sin(θ))

Solution

3 (1 + sin (θ)) = 3 (1 - sin (θ))

Solution

θ = 2 πn, θ = π + 2 πn

Degrees

θ = 0^{\circ} + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

3 (1 + sin (θ)) = 3 (1 - sin (θ))

Solve by substitution

3 (1 + sin (θ)) = 3 (1 - sin (θ))

Let:

sin (θ) = u

3 (1 + u) = 3 (1 - u)

3 (1 + u) = 3 (1 - u) : u = 0

3 (1 + u) = 3 (1 - u)

Expand

3 (1 + u) : 3 + 3 u

3 (1 + u)

Apply the distributive law:

a (b + c) = ab + a c

a = 3, b = 1, c = u

= 3 \cdot 1 + 3 u

Multiply the numbers:

3 \cdot 1 = 3

= 3 + 3 u

Expand

3 (1 - u) : 3 - 3 u

3 (1 - u)

Apply the distributive law:

a (b - c) = ab - a c

a = 3, b = 1, c = u

= 3 \cdot 1 - 3 u

Multiply the numbers:

3 \cdot 1 = 3

= 3 - 3 u

3 + 3 u = 3 - 3 u

Move

3

to the right side

3 + 3 u = 3 - 3 u

Subtract

3

from both sides

3 + 3 u - 3 = 3 - 3 u - 3

Simplify

3 u = - 3 u

3 u = - 3 u

Move

3 u

to the left side

3 u = - 3 u

Add

3 u

to both sides

3 u + 3 u = - 3 u + 3 u

Simplify

6 u = 0

6 u = 0

Divide both sides by

6

6 u = 0

Divide both sides by

6

\frac{6 u}{6} = \frac{0}{6}

Simplify

u = 0

u = 0

Substitute back

u = sin (θ)

sin (θ) = 0

sin (θ) = 0

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

sin (θ) = 0

General solutions for

sin (θ) = 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}

θ = 0 + 2 πn, θ = π + 2 πn

θ = 0 + 2 πn, θ = π + 2 πn

Solve

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn, θ = π + 2 πn

Combine all the solutions

θ = 2 πn, θ = π + 2 πn

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cos (2 x) = sin (7 0^{\circ} + x)

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2cos(θ)+5sqrt(3)=4sqrt(3)

2 cos (θ) + 53 = 43

cos(x)=-1/2 sqrt(2)

cos (x) = - \frac{1}{2} 2

sin(x)=-cos^2(x)-1,0<= x<= 2pi

sin (x) = - cos^{2} (x) - 1, 0 \leq x \leq 2 π

Frequently Asked Questions (FAQ)

What is the general solution for 3(1+sin(θ))=3(1-sin(θ)) ?
The general solution for 3(1+sin(θ))=3(1-sin(θ)) is θ=2pin,θ=pi+2pin