Maths: How Tangent Transformations Reshape the Curve

Trigonometric transformations reveal how stretching, shifting, and reflecting reshape the familiar sine, cosine, and tangent curves. Here, the function f(x) = 2tan(x/3) combines a vertical stretch by a factor of 2 with a horizontal stretch that changes the tangent's natural rhythm. The period of tan(x) is π, but dividing the input by 3 stretches it to 3π, meaning the graph completes one full cycle across a much wider interval. That single relationship — period equals π divided by the coefficient of x — governs where the curve repeats and where its asymptotes fall. Understanding this matters because tangent behaves differently from sine and cosine: it has no maximum or minimum, passing instead through zero at regular intervals while racing toward undefined values at its asymptotes. The x-intercepts occur wherever the tangent equals zero, so solving tan(x/3) = 0 means setting x/3 equal to integer multiples of π. Meanwhile, evaluating the function at a specific point, such as x = 5π/2, requires substituting carefully and recognising exact angles like 5π/6 on the unit circle. Each part connects: period shapes the domain, zeros anchor the graph, and exact values confirm the transformation.