697 Найдите синус, косинус и тангенс углов А и В треугольника АВС с прямым углом С, если: а) BC = 8, АВ = 17; б) ВС=21 АС = 20; в) BC = 1, AC = 2; г) АС = 24, AB = 25.

Рассмотрим прямоугольный треугольник ABC с прямым углом C.

a) Дано: BC = 8, AB = 17

Найдём AC по теореме Пифагора: $$AC = \sqrt{AB^2 - BC^2} = \sqrt{17^2 - 8^2} = \sqrt{289 - 64} = \sqrt{225} = 15$$

Тогда:

• $$\sin A = \frac{BC}{AB} = \frac{8}{17}$$
• $$\cos A = \frac{AC}{AB} = \frac{15}{17}$$
• $$\tan A = \frac{BC}{AC} = \frac{8}{15}$$
• $$\sin B = \frac{AC}{AB} = \frac{15}{17}$$
• $$\cos B = \frac{BC}{AB} = \frac{8}{17}$$
• $$\tan B = \frac{AC}{BC} = \frac{15}{8}$$

б) Дано: BC = 21, AC = 20

Найдём AB по теореме Пифагора: $$AB = \sqrt{AC^2 + BC^2} = \sqrt{20^2 + 21^2} = \sqrt{400 + 441} = \sqrt{841} = 29$$

Тогда:

• $$\sin A = \frac{BC}{AB} = \frac{21}{29}$$
• $$\cos A = \frac{AC}{AB} = \frac{20}{29}$$
• $$\tan A = \frac{BC}{AC} = \frac{21}{20}$$
• $$\sin B = \frac{AC}{AB} = \frac{20}{29}$$
• $$\cos B = \frac{BC}{AB} = \frac{21}{29}$$
• $$\tan B = \frac{AC}{BC} = \frac{20}{21}$$

в) Дано: BC = 1, AC = 2

Найдём AB по теореме Пифагора: $$AB = \sqrt{AC^2 + BC^2} = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5}$$

Тогда:

• $$\sin A = \frac{BC}{AB} = \frac{1}{\sqrt{5}} = \frac{\sqrt{5}}{5}$$
• $$\cos A = \frac{AC}{AB} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5}$$
• $$\tan A = \frac{BC}{AC} = \frac{1}{2}$$
• $$\sin B = \frac{AC}{AB} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5}$$
• $$\cos B = \frac{BC}{AB} = \frac{1}{\sqrt{5}} = \frac{\sqrt{5}}{5}$$
• $$\tan B = \frac{AC}{BC} = \frac{2}{1} = 2$$

г) Дано: AC = 24, AB = 25

Найдём BC по теореме Пифагора: $$BC = \sqrt{AB^2 - AC^2} = \sqrt{25^2 - 24^2} = \sqrt{625 - 576} = \sqrt{49} = 7$$

Тогда:

• $$\sin A = \frac{BC}{AB} = \frac{7}{25}$$
• $$\cos A = \frac{AC}{AB} = \frac{24}{25}$$
• $$\tan A = \frac{BC}{AC} = \frac{7}{24}$$
• $$\sin B = \frac{AC}{AB} = \frac{24}{25}$$
• $$\cos B = \frac{BC}{AB} = \frac{7}{25}$$
• $$\tan B = \frac{AC}{BC} = \frac{24}{7}$$

Ответ: а) sin A = 8/17, cos A = 15/17, tan A = 8/15, sin B = 15/17, cos B = 8/17, tan B = 15/8; б) sin A = 21/29, cos A = 20/29, tan A = 21/20, sin B = 20/29, cos B = 21/29, tan B = 20/21; в) sin A = √5/5, cos A = 2√5/5, tan A = 1/2, sin B = 2√5/5, cos B = √5/5, tan B = 2; г) sin A = 7/25, cos A = 24/25, tan A = 7/24, sin B = 24/25, cos B = 7/25, tan B = 24/7