Problem

In a right triangle, \(α=30°\), hypotenuse \(c=10\), side \(a\) is opposite \(α\), and \(β\) is the other acute angle. Use complementary angles and \(30-60-90\) side ratios to determine \(β\), \(a\), and \(b\), then report the complete solution set.

Neutral right-triangle reference stating alpha is 30 degrees, hypotenuse c is 10, a is opposite alpha, and b is adjacent alpha, without solved dimensions or a scaled triangle.

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What this problem is really about

A complete triangle solution must keep every named side tied to its angle role. We’ll find the complementary acute angle, use sine for the opposite leg and cosine for the adjacent leg, retain exact special-triangle forms, and check the side set with the Pythagorean theorem.

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β = 60°, a = 5√3, b = 5, c = 10.

β = 60°, a = 1/2, b = √3/2, c = 1.

α = 45°, β = 45°, a = b = 5√2, c = 10.

β = 60°, a = 5, b = 5√3, c = 10.

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In a right triangle, \(α=30°\), hypotenuse \(c=10\), side \(a\) is opposite \(α\), and \(β\) is the other acute angle. Use complementary angles and \(30-60-90\) side ratios to determine \(β\), \(a\), and \(b\), then report the complete solution set.

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