а) Зная, что 15 cos 2t + 8 sin t = 9 и 1 < t< 3, вычислите tg t;

1. Дано:

15cos2t + 8sint = 9; 1 < t < 3; или 0 < t <π; t принадлежит первой или второй четверти; sint> 0.

2.

15cos2t + 8sint = 9; 15 (1 - 2sin^2t) + 8sint = 9; 15 - 30sin^2t + 8sint - 9 = 0; 6 - 30sin^2t + 8sint = 0; 30sin^2t - 8sint - 6 = 0; 15sin^2t - 4sint - 3 = 0; D/4 = 2^2 + 15 * 3 = 4 + 45 = 49 = 7^2;

1) sint = (2 - 7) / 15 = - 5/15 = - 1/3 < 0 - не подходит;

2) sint = (2 + 7) / 15 = 9/15 = 3/5.

3.

cost = ±√ (1 - sin^2t); cost = ±√ (1 - (3/5) ^2) = ±√ (1 - 9/25) = ±√ (16/25) = ±4/5; tgt = sint/cost; tgt = 3/5 : (±4/5) = ±3/5 * 5/4 = ±3/4.

Ответ: ±3/4.