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Popular >

0=-6sin^2(θ)+3sin(θ)+2

Solution

0 = - 6 sin^{2} (θ) + 3 sin (θ) + 2

θ = - 0.38888 \dots + 2 πn, θ = π + 0.38888 \dots + 2 πn, θ = 1.07408 \dots + 2 πn, θ = π - 1.07408 \dots + 2 πn

Degrees

θ = - 22.28121 \dots^{\circ} + 36 0^{\circ} n, θ = 202.28121 \dots^{\circ} + 36 0^{\circ} n, θ = 61.54034 \dots^{\circ} + 36 0^{\circ} n, θ = 118.45965 \dots^{\circ} + 36 0^{\circ} n

Solution steps

0 = - 6 sin^{2} (θ) + 3 sin (θ) + 2

Switch sides

- 6 sin^{2} (θ) + 3 sin (θ) + 2 = 0

Solve by substitution

- 6 sin^{2} (θ) + 3 sin (θ) + 2 = 0

Let:

sin (θ) = u

- 6 u^{2} + 3 u + 2 = 0

- 6 u^{2} + 3 u + 2 = 0 : u = - \frac{- 3 + 57}{12}, u = \frac{3 + 57}{12}

- 6 u^{2} + 3 u + 2 = 0

Solve with the quadratic formula

- 6 u^{2} + 3 u + 2 = 0

Quadratic Equation Formula:

For

a = - 6, b = 3, c = 2

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 ( - 6 ) \cdot 2}{2 ( - 6 )}

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 ( - 6 ) \cdot 2}{2 ( - 6 )}

3^{2} - 4 (- 6) \cdot 2 = 57

3^{2} - 4 (- 6) \cdot 2

Apply rule

- (- a) = a

= 3^{2} + 4 \cdot 6 \cdot 2

Multiply the numbers:

4 \cdot 6 \cdot 2 = 48

= 3^{2} + 48

3^{2} = 9

= 9 + 48

Add the numbers:

9 + 48 = 57

= 57

u_{1, 2} = \frac{- 3 \pm 57}{2 ( - 6 )}

Separate the solutions

u_{1} = \frac{- 3 + 57}{2 ( - 6 )}, u_{2} = \frac{- 3 - 57}{2 ( - 6 )}

u = \frac{- 3 + 57}{2 ( - 6 )} : - \frac{- 3 + 57}{12}

\frac{- 3 + 57}{2 ( - 6 )}

Remove parentheses:

(- a) = - a

= \frac{- 3 + 57}{- 2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{- 3 + 57}{- 12}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{- 3 + 57}{12}

u = \frac{- 3 - 57}{2 ( - 6 )} : \frac{3 + 57}{12}

\frac{- 3 - 57}{2 ( - 6 )}

Remove parentheses:

(- a) = - a

= \frac{- 3 - 57}{- 2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{- 3 - 57}{- 12}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

- 3 - 57 = - (3 + 57)

= \frac{3 + 57}{12}

The solutions to the quadratic equation are:

u = - \frac{- 3 + 57}{12}, u = \frac{3 + 57}{12}

Substitute back

u = sin (θ)

sin (θ) = - \frac{- 3 + 57}{12}, sin (θ) = \frac{3 + 57}{12}

sin (θ) = - \frac{- 3 + 57}{12}, sin (θ) = \frac{3 + 57}{12}

sin (θ) = - \frac{- 3 + 57}{12} : θ = arcsin (- \frac{- 3 + 57}{12}) + 2 πn, θ = π + arcsin (\frac{- 3 + 57}{12}) + 2 πn

sin (θ) = - \frac{- 3 + 57}{12}

Apply trig inverse properties

sin (θ) = - \frac{- 3 + 57}{12}

General solutions for

sin (θ) = - \frac{- 3 + 57}{12}

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

θ = arcsin (- \frac{- 3 + 57}{12}) + 2 πn, θ = π + arcsin (\frac{- 3 + 57}{12}) + 2 πn

θ = arcsin (- \frac{- 3 + 57}{12}) + 2 πn, θ = π + arcsin (\frac{- 3 + 57}{12}) + 2 πn

sin (θ) = \frac{3 + 57}{12} : θ = arcsin (\frac{3 + 57}{12}) + 2 πn, θ = π - arcsin (\frac{3 + 57}{12}) + 2 πn

sin (θ) = \frac{3 + 57}{12}

Apply trig inverse properties

sin (θ) = \frac{3 + 57}{12}

General solutions for

sin (θ) = \frac{3 + 57}{12}

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

θ = arcsin (\frac{3 + 57}{12}) + 2 πn, θ = π - arcsin (\frac{3 + 57}{12}) + 2 πn

θ = arcsin (\frac{3 + 57}{12}) + 2 πn, θ = π - arcsin (\frac{3 + 57}{12}) + 2 πn

Combine all the solutions

θ = arcsin (- \frac{- 3 + 57}{12}) + 2 πn, θ = π + arcsin (\frac{- 3 + 57}{12}) + 2 πn, θ = arcsin (\frac{3 + 57}{12}) + 2 πn, θ = π - arcsin (\frac{3 + 57}{12}) + 2 πn

Show solutions in decimal form

θ = - 0.38888 \dots + 2 πn, θ = π + 0.38888 \dots + 2 πn, θ = 1.07408 \dots + 2 πn, θ = π - 1.07408 \dots + 2 πn

sin (2 x + \frac{3}{π}) = \frac{1}{2}

3 cot (\frac{x}{3}) + 1 = 0

sin (x) = 0.1115

arccos (x) = 11555

sin (A) = - \frac{1}{2}