What is the general solution for sin(θ)=-(sqrt(68))/9 ?

The general solution for sin(θ)=-(sqrt(68))/9 is θ=-1.15860…+2pin,θ=pi+1.15860…+2pin

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sin(θ)=-(sqrt(68))/9

sin (θ) = - \frac{68}{9}

θ = - 1.15860 \dots + 2 πn, θ = π + 1.15860 \dots + 2 πn

Degrees

θ = - 66.38325 \dots^{\circ} + 36 0^{\circ} n, θ = 246.38325 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin (θ) = - \frac{68}{9}

68 = 217

68

Prime factorization of

68 : 2^{2} \cdot 17

68

68

divides by

268 = 34 \cdot 2

= 2 \cdot 34

34

divides by

234 = 17 \cdot 2

= 2 \cdot 2 \cdot 17

2, 17

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 17

= 2^{2} \cdot 17

= 2^{2} \cdot 17

Apply radical rule:

n ab = n a n b

= 17 2^{2}

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 217

sin (θ) = - \frac{2 17}{9}

Apply trig inverse properties

sin (θ) = - \frac{2 17}{9}

General solutions for

sin (θ) = - \frac{2 17}{9}

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

θ = arcsin (- \frac{2 17}{9}) + 2 πn, θ = π + arcsin (\frac{2 17}{9}) + 2 πn

θ = arcsin (- \frac{2 17}{9}) + 2 πn, θ = π + arcsin (\frac{2 17}{9}) + 2 πn

Show solutions in decimal form

θ = - 1.15860 \dots + 2 πn, θ = π + 1.15860 \dots + 2 πn

sec (θ) = 6

tan (2 x - \frac{π}{3}) = - 3

3 sin^{2} (2 θ) + 8 cos (2 θ) = 0

cos (2 x - \frac{π}{3}) = \frac{1}{2}

sin (θ) = \frac{7}{11}

What is the general solution for sin(θ)=-(sqrt(68))/9 ?
The general solution for sin(θ)=-(sqrt(68))/9 is θ=-1.15860…+2pin,θ=pi+1.15860…+2pin