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Question

solve: tan (theta/2) - sin(theta)=0

Answer 1

The trigonometric equation is tan(θ/2) - sin(θ) = 0.

Half - angle formula : sin θ = 2 tan (θ/2) / (1 + tan² (θ/2)).

tan(θ/2) - [ 2 tan (θ/2) / (1 + tan² (θ/2)) ] = 0

tan(θ/2)(1 + tan² (θ/2)) - 2 tan (θ/2) = 0

tan(θ/2)[ (1 + tan² (θ/2)) - 2 ] = 0

tan(θ/2)[ (tan² (θ/2)) - 1 ] = 0

⇒ tan(θ/2) = 0  and  tan² (θ/2) - 1 = 0

• tan(θ/2) = 0.

tan (θ/2) = tan 0.

The genaral solution of tan(θ) = tan(α) is θ = nπ + α, where n is an integer.

θ/2 = nπ + 0

⇒ θ = 2nπ.

• tan² (θ/2) - 1 = 0

tan² (θ/2) = 1

tan (θ/2) = ± 1

tan (θ/2) = - 1 and tan (θ/2) = + 1

tan (θ/2) = tan(3π/4) and tan (θ/2) = tan(π/4)

θ/2 = nπ + (3π/4)  and  θ/2 = nπ + (π/4)

⇒ θ = 2nπ + (3π/2)  and  θ = 2nπ + (π/2).

Therefore, the solutions are θ = 2nπ, θ = 2nπ + (3π/2) and θ = 2nπ + (π/2), where n is an integer.