Equation Involving Trigonometric Identities

Question

Solve this equation for 0 ≤ θ < 2π:

tanθsinθ - 4tanθ = -3tanθ

Answer 1

It's tempting to do the following:

tanθsinθ - 4tanθ = -3tanθ

tanθsinθ = tanθ

sinθ = 1

On the domain specified in the question, sinθ = 1 when θ = π/2.

^ This is tempting, but it's not complete! In dividing out tanθ from both sides, I might potentially lose information, or I actually might have an undefined solution if I divide everything by tanθ if this is actually 0 and I don't realize it, so I'll try this again with a slightly different approach:

tanθsinθ - 4tanθ = -3 tanθ

tanθsinθ - tanθ = 0

tanθ(sinθ - 1) = 0

By the zero-product property:

tanθ = 0 and sinθ - 1 = 0

On the domain specified in the question, tanθ = 0 when θ = 0 and when θ = π. (Don't include 2π since the domain doesn't include 2π; watch the inequality signs.)

For sinθ - 1 = 0, this becomes sinθ = 1. Look familiar?

On the domain specified in the question, sinθ = 1 when θ = π/2.

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So far, we have θ = 0, π/2, and π. Plug them into the original equation to check these solutions:

tanθsinθ - 4tanθ = -3 tanθ

For θ = 0, since tan(0) = 0, the whole equation reduces to 0 = 0, a true statement.

For θ = π/2, trying to plug in tan(π/2) into the original equation will be problematic since this value is undefined (look at the graph of y = tan θ, and you'll see an asymptote at θ = π/2). This is extraneous!

For θ = π, since tan(π) = 0, the whole equation reduces to 0 = 0, a true statement.

Final answer: On the domain specified in the question, θ = 0 and θ = π.