Problem

The similar \(3-4-5\) and \(6-8-10\) right triangles share reference angle \(θ\), with opposite, adjacent, and hypotenuse corresponding in that order. Calculate \(\sin(θ)\), \(\cos(θ)\), and \(\tan(θ)\), then verify that scaling leaves each ratio unchanged.

Prompt diagram of similar right triangles with the same reference angle theta. In the smaller triangle, opposite = 3, adjacent = 4, hypotenuse = 5; in the scale-factor-2 triangle, the corresponding sides are 6, 8, and 10. The trig ratios are withheld.

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What this problem is really about

Similar right triangles preserve trig ratios because scaling multiplies both terms of each fraction equally. We’ll align opposite, adjacent, and hypotenuse roles, compute sine, cosine, and tangent from each triangle, simplify the larger ratios, and compare the pairs to confirm angle-based invariance.

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sin(θ) = 3/5, cos(θ) = 3/5, tan(θ) = 3/5

sin(θ) = 6/5, cos(θ) = 8/5, tan(θ) = 6/4

sin(θ) = 3/5, cos(θ) = 4/5, tan(θ) = 3/4

sin(θ) = 3/5, cos(θ) = 4/5, tan(θ) = 4/3

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The similar \(3-4-5\) and \(6-8-10\) right triangles share reference angle \(θ\), with opposite, adjacent, and hypotenuse corresponding in that order. Calculate \(\sin(θ)\), \(\cos(θ)\), and \(\tan(θ)\), then verify that scaling leaves each ratio unchanged.

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