4. Найдите синус, косинус, тангенс углов А и В прямоугольного треугольника АВС, если: a) AC3, AB5; 6) AC-10, ВС-8; в) ВС 3√3, AB-6v2.

Ответ: a) sin A = 4/5, cos A = 3/5, tg A = 4/3, sin B = 3/5, cos B = 4/5, tg B = 3/4; б) sin A = 8/\(\sqrt{164}\), cos A = 10/\(\sqrt{164}\), tg A = 8/10, sin B = 10/\(\sqrt{164}\), cos B = 8/\(\sqrt{164}\), tg B = 10/8; в) sin A = 1/2, cos A = \(\sqrt{3}\)/2, tg A = \(\sqrt{3}\)/3, sin B = \(\sqrt{3}\)/2, cos B = 1/2, tg B = \(\sqrt{3}\)

Решение:

a) Дано: AC = 3, AB = 5.

• Найдем BC по теореме Пифагора:

\[BC = \sqrt{AB^2 - AC^2} = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4\]

• sin A = BC/AB = 4/5
• cos A = AC/AB = 3/5
• tg A = BC/AC = 4/3
• sin B = AC/AB = 3/5
• cos B = BC/AB = 4/5
• tg B = AC/BC = 3/4

б) Дано: AC = 10, BC = 8.

• Найдем AB по теореме Пифагора:

\[AB = \sqrt{AC^2 + BC^2} = \sqrt{10^2 + 8^2} = \sqrt{100 + 64} = \sqrt{164}\]

• sin A = BC/AB = 8/\(\sqrt{164}\)
• cos A = AC/AB = 10/\(\sqrt{164}\)
• tg A = BC/AC = 8/10
• sin B = AC/AB = 10/\(\sqrt{164}\)
• cos B = BC/AB = 8/\(\sqrt{164}\)
• tg B = AC/BC = 10/8

в) Дано: BC = 3\(\sqrt{3}\), AB = 6\(\sqrt{2}\)

• Найдем AC по теореме Пифагора:

\[AC = \sqrt{AB^2 - BC^2} = \sqrt{(6\sqrt{2})^2 - (3\sqrt{3})^2} = \sqrt{72 - 27} = \sqrt{45} = 3\sqrt{5}\]

• sin A = BC/AB = (3\(\sqrt{3}\))/(6\(\sqrt{2}\)) = \(\sqrt{6}\)/4
• cos A = AC/AB = (3\(\sqrt{5}\))/(6\(\sqrt{2}\)) = \(\sqrt{10}\)/4
• tg A = BC/AC = (3\(\sqrt{3}\))/(3\(\sqrt{5}\)) = \(\sqrt{15}\)/5
• sin B = AC/AB = (3\(\sqrt{5}\))/(6\(\sqrt{2}\)) = \(\sqrt{10}\)/4
• cos B = BC/AB = (3\(\sqrt{3}\))/(6\(\sqrt{2}\)) = \(\sqrt{6}\)/4
• tg B = AC/BC = (3\(\sqrt{5}\))/(3\(\sqrt{3}\)) = \(\sqrt{15}\)/3

Ответ: a) sin A = 4/5, cos A = 3/5, tg A = 4/3, sin B = 3/5, cos B = 4/5, tg B = 3/4; б) sin A = 8/\(\sqrt{164}\), cos A = 10/\(\sqrt{164}\), tg A = 8/10, sin B = 10/\(\sqrt{164}\), cos B = 8/\(\sqrt{164}\), tg B = 10/8; в) sin A = 1/2, cos A = \(\sqrt{3}\)/2, tg A = \(\sqrt{3}\)/3, sin B = \(\sqrt{3}\)/2, cos B = 1/2, tg B = \(\sqrt{3}\)