What is the general solution for -sin(a)-1=3sin(a)+2 ?

The general solution for -sin(a)-1=3sin(a)+2 is a=-0.84806…+2pin,a=pi+0.84806…+2pin

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-sin(a)-1=3sin(a)+2

- sin (a) - 1 = 3 sin (a) + 2

a = - 0.84806 \dots + 2 πn, a = π + 0.84806 \dots + 2 πn

Degrees

a = - 48.59037 \dots^{\circ} + 36 0^{\circ} n, a = 228.59037 \dots^{\circ} + 36 0^{\circ} n

Solution steps

- sin (a) - 1 = 3 sin (a) + 2

Solve by substitution

- sin (a) - 1 = 3 sin (a) + 2

Let:

sin (a) = u

- u - 1 = 3 u + 2

- u - 1 = 3 u + 2 : u = - \frac{3}{4}

- u - 1 = 3 u + 2

Move

1

to the right side

- u - 1 = 3 u + 2

Add

1

to both sides

- u - 1 + 1 = 3 u + 2 + 1

Simplify

- u = 3 u + 3

- u = 3 u + 3

Move

3 u

to the left side

- u = 3 u + 3

Subtract

3 u

from both sides

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

Simplify

- 4 u = 3

- 4 u = 3

Divide both sides by

- 4

- 4 u = 3

Divide both sides by

- 4

\frac{- 4 u}{- 4} = \frac{3}{- 4}

Simplify

u = - \frac{3}{4}

u = - \frac{3}{4}

Substitute back

u = sin (a)

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

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

sin (a) = - \frac{3}{4} : a = arcsin (- \frac{3}{4}) + 2 πn, a = π + arcsin (\frac{3}{4}) + 2 πn

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

Apply trig inverse properties

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

General solutions for

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

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

a = arcsin (- \frac{3}{4}) + 2 πn, a = π + arcsin (\frac{3}{4}) + 2 πn

a = arcsin (- \frac{3}{4}) + 2 πn, a = π + arcsin (\frac{3}{4}) + 2 πn

Combine all the solutions

a = arcsin (- \frac{3}{4}) + 2 πn, a = π + arcsin (\frac{3}{4}) + 2 πn

Show solutions in decimal form

a = - 0.84806 \dots + 2 πn, a = π + 0.84806 \dots + 2 πn

tan (x) = 7

cot (x) (tan (x) - 3) = 0

cos^{2} (x) - 2 cos (x) + 1 = 0

(cos (θ) + sin (θ))^{2} = \frac{3}{2}

- 2 sin^{2} (θ) + 1 = cos (θ)

What is the general solution for -sin(a)-1=3sin(a)+2 ?
The general solution for -sin(a)-1=3sin(a)+2 is a=-0.84806…+2pin,a=pi+0.84806…+2pin