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Popular >

sin^2(θ)(1+csc^2(θ))=k

Solution

sin^{2} (θ) (1 + csc^{2} (θ)) = k

Solution

θ = arcsin (k - 1) + 2 πn, θ = π + arcsin (- k - 1) + 2 πn, θ = arcsin (- k - 1) + 2 πn, θ = π + arcsin (k - 1) + 2 πn

Solution steps

sin^{2} (θ) (1 + csc^{2} (θ)) = k

Subtract

k

from both sides

sin^{2} (θ) (1 + csc^{2} (θ)) - k = 0

Rewrite using trig identities

1 + sin^{2} (θ) - k = 0

Solve by substitution

sin (θ) = k - 1, sin (θ) = - k - 1

sin (θ) = k - 1 : θ = arcsin (k - 1) + 2 πn, θ = π + arcsin (- k - 1) + 2 πn

sin (θ) = - k - 1 : θ = arcsin (- k - 1) + 2 πn, θ = π + arcsin (k - 1) + 2 πn

Combine all the solutions

θ = arcsin (k - 1) + 2 πn, θ = π + arcsin (- k - 1) + 2 πn, θ = arcsin (- k - 1) + 2 πn, θ = π + arcsin (k - 1) + 2 πn

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