Решение задач на применение основного тригонометрического тождества

Основное тригонометрическое тождество: $$sin^2(A) + cos^2(A) = 1$$

1. Дано: $$sin(A) = \frac{\sqrt{21}}{5}$$. Найти: $$cos(A)$$.

   Решение: $$cos^2(A) = 1 - sin^2(A) = 1 - (\frac{\sqrt{21}}{5})^2 = 1 - \frac{21}{25} = \frac{4}{25}$$

   $$cos(A) = \sqrt{\frac{4}{25}} = \frac{2}{5}$$

   Ответ: $$cos(A) = \frac{2}{5}$$

2. Дано: $$sin(A) = \frac{3\sqrt{11}}{10}$$. Найти: $$cos(A)$$.

   Решение: $$cos^2(A) = 1 - sin^2(A) = 1 - (\frac{3\sqrt{11}}{10})^2 = 1 - \frac{9 \cdot 11}{100} = 1 - \frac{99}{100} = \frac{1}{100}$$

   $$cos(A) = \sqrt{\frac{1}{100}} = \frac{1}{10}$$

   Ответ: $$cos(A) = \frac{1}{10}$$

3. Дано: $$sin(A) = \frac{3}{5}$$. Найти: $$cos(A)$$.

   Решение: $$cos^2(A) = 1 - sin^2(A) = 1 - (\frac{3}{5})^2 = 1 - \frac{9}{25} = \frac{16}{25}$$

   $$cos(A) = \sqrt{\frac{16}{25}} = \frac{4}{5}$$

   Ответ: $$cos(A) = \frac{4}{5}$$

4. Дано: $$cos(A) = \frac{\sqrt{91}}{10}$$. Найти: $$sin(A)$$.

   Решение: $$sin^2(A) = 1 - cos^2(A) = 1 - (\frac{\sqrt{91}}{10})^2 = 1 - \frac{91}{100} = \frac{9}{100}$$

   $$sin(A) = \sqrt{\frac{9}{100}} = \frac{3}{10}$$

   Ответ: $$sin(A) = \frac{3}{10}$$

5. Дано: $$cos(A) = \frac{4}{5}$$. Найти: $$sin(A)$$.

   Решение: $$sin^2(A) = 1 - cos^2(A) = 1 - (\frac{4}{5})^2 = 1 - \frac{16}{25} = \frac{9}{25}$$

   $$sin(A) = \sqrt{\frac{9}{25}} = \frac{3}{5}$$

   Ответ: $$sin(A) = \frac{3}{5}$$

6. Дано: $$cos(A) = \frac{3\sqrt{7}}{8}$$. Найти: $$sin(A)$$.

   Решение: $$sin^2(A) = 1 - cos^2(A) = 1 - (\frac{3\sqrt{7}}{8})^2 = 1 - \frac{9 \cdot 7}{64} = 1 - \frac{63}{64} = \frac{1}{64}$$

   $$sin(A) = \sqrt{\frac{1}{64}} = \frac{1}{8}$$

   Ответ: $$sin(A) = \frac{1}{8}$$