13 AB = 5 BC sin ∠B-? cos ∠B-? tg ∠B-? ctg ∠B-? B C A

Ответ: \(\sin B = \frac{24}{25}, \cos B = \frac{7}{25}, tg B = \frac{24}{7}, ctg B = \frac{7}{24}\)

Разбираемся:

1. В прямоугольном треугольнике ABC, \(\angle C = 90^\circ\), AB = 5BC (дано).
2. Пусть BC = x, тогда AB = 5x.
3. По теореме Пифагора, \(AC^2 = AB^2 - BC^2 = (5x)^2 - x^2 = 25x^2 - x^2 = 24x^2\), значит, \(AC = \sqrt{24x^2} = x\sqrt{24}\).
4. Теперь найдем тригонометрические функции угла B:

• \(\sin B = \frac{AC}{AB} = \frac{x\sqrt{24}}{5x} = \frac{\sqrt{24}}{5} = \frac{2\sqrt{6}}{5}\)
• \(\cos B = \frac{BC}{AB} = \frac{x}{5x} = \frac{1}{5}\)
• \(tg B = \frac{AC}{BC} = \frac{x\sqrt{24}}{x} = \sqrt{24} = 2\sqrt{6}\)
• \(ctg B = \frac{BC}{AC} = \frac{x}{x\sqrt{24}} = \frac{1}{\sqrt{24}} = \frac{1}{2\sqrt{6}}\)

Ответ: \(\sin B = \frac{2\sqrt{6}}{5}, \cos B = \frac{1}{5}, tg B = 2\sqrt{6}, ctg B = \frac{1}{2\sqrt{6}}\)