4. Найдите синус, косинус, тангенс углов А и В прямоугольного треугольника АВС, если: а) АС = 4,AB = 5; 6) AC=15, BC=8; в) ВС = 6√3, AB=9√2.

4. Рассмотрим прямоугольный треугольник ABC, где угол C = 90°.

а) Дано: AC = 4, AB = 5. Тогда BC = √(AB² - AC²) = √(5² - 4²) = √(25 - 16) = √9 = 3.

• sin A = BC/AB = 3/5 = 0.6
• cos A = AC/AB = 4/5 = 0.8
• tg A = BC/AC = 3/4 = 0.75
• sin B = AC/AB = 4/5 = 0.8
• cos B = BC/AB = 3/5 = 0.6
• tg B = AC/BC = 4/3 = 1.333

б) Дано: AC = 15, BC = 8. Тогда AB = √(AC² + BC²) = √(15² + 8²) = √(225 + 64) = √289 = 17.

• sin A = BC/AB = 8/17 = 0.471
• cos A = AC/AB = 15/17 = 0.882
• tg A = BC/AC = 8/15 = 0.533
• sin B = AC/AB = 15/17 = 0.882
• cos B = BC/AB = 8/17 = 0.471
• tg B = AC/BC = 15/8 = 1.875

в) Дано: BC = 6√3, AB = 9√2. Тогда AC = √(AB² - BC²) = √((9√2)² - (6√3)²) = √(162 - 108) = √54 = 3√6.

• sin A = BC/AB = (6√3) / (9√2) = (2√3) / (3√2) = (2√6) / 6 = √6 / 3 = 0.816
• cos A = AC/AB = (3√6) / (9√2) = (√6) / (3√2) = √3 / 3 = 0.577
• tg A = BC/AC = (6√3) / (3√6) = 2√3 / √6 = 2√18 / 6 = 2 * 3√2 / 6 = √2 = 1.414
• sin B = AC/AB = (3√6) / (9√2) = (√6) / (3√2) = √3 / 3 = 0.577
• cos B = BC/AB = (6√3) / (9√2) = (2√3) / (3√2) = (2√6) / 6 = √6 / 3 = 0.816
• tg B = AC/BC = (3√6) / (6√3) = √6 / (2√3) = √2 / 2 = 0.707

Ответ: a) sin A = 0.6, cos A = 0.8, tg A = 0.75, sin B = 0.8, cos B = 0.6, tg B = 1.333; б) sin A = 0.471, cos A = 0.882, tg A = 0.533, sin B = 0.882, cos B = 0.471, tg B = 1.875; в) sin A = 0.816, cos A = 0.577, tg A = 1.414, sin B = 0.577, cos B = 0.816, tg B = 0.707