Глава V. § 31. Формулы приведения — с. 160
527. Упростить выражение:
1) (ctg (π/2 − α) − tg (π + α) + sin (3π/2 − α))/cos (π + α);
2) (sin (π − α) + cos (π/2 + α) + ctg (π − α))/tg (3π/2 − α).
1) (ctg (π/2 − α) − tg (π + α) + sin (3π/2 − α))/cos (π + α);
2) (sin (π − α) + cos (π/2 + α) + ctg (π − α))/tg (3π/2 − α).
1) ctg (π/2 − α) = tg α, tg (π + α) = tg α, sin (3π/2 − α) = −cos α, cos (π + α) = −cos α. Числитель: tg α − tg α − cos α = −cos α. Дробь равна (−cos α)/(−cos α) = 1.
2) sin (π − α) = sin α, cos (π/2 + α) = −sin α, ctg (π − α) = −ctg α, tg (3π/2 − α) = ctg α. Числитель: sin α − sin α − ctg α = −ctg α. Дробь равна −ctg α/ctg α = −1.
Ответ: 1) 1; 2) −1.
2) sin (π − α) = sin α, cos (π/2 + α) = −sin α, ctg (π − α) = −ctg α, tg (3π/2 − α) = ctg α. Числитель: sin α − sin α − ctg α = −ctg α. Дробь равна −ctg α/ctg α = −1.
Ответ: 1) 1; 2) −1.
528. Упростить выражение:
1) sin (3π/2 + α)/ctg (2π − α) · tg (π/2 + α)/sin (π + α);
2) (sin² (π + α) + sin² (π/2 + α))/cos (3π/2 + α) · ctg (3π/2 − α).
1) sin (3π/2 + α)/ctg (2π − α) · tg (π/2 + α)/sin (π + α);
2) (sin² (π + α) + sin² (π/2 + α))/cos (3π/2 + α) · ctg (3π/2 − α).
1) sin (3π/2 + α) = −cos α, ctg (2π − α) = −ctg α, tg (π/2 + α) = −ctg α, sin (π + α) = −sin α. Получаем (−cos α)/(−ctg α) · (−ctg α)/(−sin α) = (cos α/ctg α) · (ctg α/sin α) = cos α/sin α = ctg α.
2) sin² (π + α) = sin² α, sin² (π/2 + α) = cos² α, их сумма равна 1; cos (3π/2 + α) = sin α, ctg (3π/2 − α) = tg α. Выражение равно (1/sin α) · tg α = (1/sin α) · (sin α/cos α) = 1/cos α.
Ответ: 1) ctg α; 2) 1/cos α.
2) sin² (π + α) = sin² α, sin² (π/2 + α) = cos² α, их сумма равна 1; cos (3π/2 + α) = sin α, ctg (3π/2 − α) = tg α. Выражение равно (1/sin α) · tg α = (1/sin α) · (sin α/cos α) = 1/cos α.
Ответ: 1) ctg α; 2) 1/cos α.
529. Вычислить:
1) cos 750°;
2) sin 1140°;
3) tg 405°;
4) cos 840°;
5) sin 47π/6;
6) tg 25π/4;
7) ctg 27π/4;
8) cos 21π/4.
1) cos 750°;
2) sin 1140°;
3) tg 405°;
4) cos 840°;
5) sin 47π/6;
6) tg 25π/4;
7) ctg 27π/4;
8) cos 21π/4.
Отбрасываем целое число периодов (360° или 2π для sin, cos; 180° или π для tg, ctg).
1) cos 750° = cos (2 · 360° + 30°) = cos 30° = √3/2.
2) sin 1140° = sin (3 · 360° + 60°) = sin 60° = √3/2.
3) tg 405° = tg (360° + 45°) = tg 45° = 1.
4) cos 840° = cos (2 · 360° + 120°) = cos 120° = −1/2.
5) sin 47π/6 = sin (6π + 11π/6) = sin 11π/6 = sin (2π − π/6) = −1/2.
6) tg 25π/4 = tg (6π + π/4) = tg π/4 = 1.
7) ctg 27π/4 = ctg (6π + 3π/4) = ctg 3π/4 = −1.
8) cos 21π/4 = cos (4π + 5π/4) = cos 5π/4 = −√2/2.
Ответ: 1) √3/2; 2) √3/2; 3) 1; 4) −1/2; 5) −1/2; 6) 1; 7) −1; 8) −√2/2.
1) cos 750° = cos (2 · 360° + 30°) = cos 30° = √3/2.
2) sin 1140° = sin (3 · 360° + 60°) = sin 60° = √3/2.
3) tg 405° = tg (360° + 45°) = tg 45° = 1.
4) cos 840° = cos (2 · 360° + 120°) = cos 120° = −1/2.
5) sin 47π/6 = sin (6π + 11π/6) = sin 11π/6 = sin (2π − π/6) = −1/2.
6) tg 25π/4 = tg (6π + π/4) = tg π/4 = 1.
7) ctg 27π/4 = ctg (6π + 3π/4) = ctg 3π/4 = −1.
8) cos 21π/4 = cos (4π + 5π/4) = cos 5π/4 = −√2/2.
Ответ: 1) √3/2; 2) √3/2; 3) 1; 4) −1/2; 5) −1/2; 6) 1; 7) −1; 8) −√2/2.
530. Найти значение выражения:
1) cos 630° − sin 1470° − ctg 1125°;
2) tg 1800° − sin 495° + cos 945°;
3) 3 cos 3660° + sin (−1560°) + cos (−450°);
4) cos 4455° − cos (−945°) + tg 1035° − ctg (−1500°).
1) cos 630° − sin 1470° − ctg 1125°;
2) tg 1800° − sin 495° + cos 945°;
3) 3 cos 3660° + sin (−1560°) + cos (−450°);
4) cos 4455° − cos (−945°) + tg 1035° − ctg (−1500°).
1) cos 630° = cos (360° + 270°) = 0; sin 1470° = sin (4 · 360° + 30°) = 1/2; ctg 1125° = ctg (6 · 180° + 45°) = 1. Значение: 0 − 1/2 − 1 = −3/2.
2) tg 1800° = tg (10 · 180°) = 0; sin 495° = sin 135° = √2/2; cos 945° = cos (2 · 360° + 225°) = −√2/2. Значение: 0 − √2/2 − √2/2 = −√2.
3) cos 3660° = cos (10 · 360° + 60°) = 1/2; sin (−1560°) = −sin 1560° = −sin (4 · 360° + 120°) = −√3/2; cos (−450°) = cos 450° = cos 90° = 0. Значение: 3/2 − √3/2 + 0 = (3 − √3)/2.
4) cos 4455° = cos (12 · 360° + 135°) = −√2/2; cos (−945°) = cos 945° = −√2/2; tg 1035° = tg (6 · 180° − 45°) = −1; ctg (−1500°) = −ctg 1500° = −ctg (8 · 180° + 60°) = −√3/3. Значение: −√2/2 + √2/2 − 1 + √3/3 = (√3 − 3)/3.
Ответ: 1) −3/2; 2) −√2; 3. (3 − √3)/2; 4. (√3 − 3)/3.
2) tg 1800° = tg (10 · 180°) = 0; sin 495° = sin 135° = √2/2; cos 945° = cos (2 · 360° + 225°) = −√2/2. Значение: 0 − √2/2 − √2/2 = −√2.
3) cos 3660° = cos (10 · 360° + 60°) = 1/2; sin (−1560°) = −sin 1560° = −sin (4 · 360° + 120°) = −√3/2; cos (−450°) = cos 450° = cos 90° = 0. Значение: 3/2 − √3/2 + 0 = (3 − √3)/2.
4) cos 4455° = cos (12 · 360° + 135°) = −√2/2; cos (−945°) = cos 945° = −√2/2; tg 1035° = tg (6 · 180° − 45°) = −1; ctg (−1500°) = −ctg 1500° = −ctg (8 · 180° + 60°) = −√3/3. Значение: −√2/2 + √2/2 − 1 + √3/3 = (√3 − 3)/3.
Ответ: 1) −3/2; 2) −√2; 3. (3 − √3)/2; 4. (√3 − 3)/3.
531. Вычислить:
1) cos 23π/4 − sin 15π/4 − ctg (−11π/2);
2) sin 25π/3 − cos (−17π/2) − tg 10π/3;
3) sin (−7π) − 2 cos 31π/3 − tg 7π/4;
4) cos (−9π) + 2 sin (−49π/6) − ctg (−21π/4).
1) cos 23π/4 − sin 15π/4 − ctg (−11π/2);
2) sin 25π/3 − cos (−17π/2) − tg 10π/3;
3) sin (−7π) − 2 cos 31π/3 − tg 7π/4;
4) cos (−9π) + 2 sin (−49π/6) − ctg (−21π/4).
1) cos 23π/4 = cos (4π + 7π/4) = cos 7π/4 = √2/2; sin 15π/4 = sin (2π + 7π/4) = −√2/2; ctg (−11π/2) = −ctg (5π + π/2) = −ctg π/2 = 0. Значение: √2/2 + √2/2 − 0 = √2.
2) sin 25π/3 = sin (8π + π/3) = √3/2; cos (−17π/2) = cos (8π + π/2) = 0; tg 10π/3 = tg (3π + π/3) = √3. Значение: √3/2 − 0 − √3 = −√3/2.
3) sin (−7π) = 0; cos 31π/3 = cos (10π + π/3) = 1/2, поэтому 2 cos 31π/3 = 1; tg 7π/4 = tg (2π − π/4) = −1. Значение: 0 − 1 + 1 = 0.
4) cos (−9π) = cos 9π = −1; sin (−49π/6) = −sin (8π + π/6) = −1/2, поэтому 2 sin (−49π/6) = −1; ctg (−21π/4) = −ctg (5π + π/4) = −1. Значение: −1 − 1 + 1 = −1.
Ответ: 1) √2; 2) −√3/2; 3) 0; 4) −1.
2) sin 25π/3 = sin (8π + π/3) = √3/2; cos (−17π/2) = cos (8π + π/2) = 0; tg 10π/3 = tg (3π + π/3) = √3. Значение: √3/2 − 0 − √3 = −√3/2.
3) sin (−7π) = 0; cos 31π/3 = cos (10π + π/3) = 1/2, поэтому 2 cos 31π/3 = 1; tg 7π/4 = tg (2π − π/4) = −1. Значение: 0 − 1 + 1 = 0.
4) cos (−9π) = cos 9π = −1; sin (−49π/6) = −sin (8π + π/6) = −1/2, поэтому 2 sin (−49π/6) = −1; ctg (−21π/4) = −ctg (5π + π/4) = −1. Значение: −1 − 1 + 1 = −1.
Ответ: 1) √2; 2) −√3/2; 3) 0; 4) −1.
532. Доказать тождество:
1) sin (π/4 + α) − cos (π/4 − α) = 0;
2) cos (π/6 − α) − sin (π/3 + α) = 0;
3) sin (3π/2 − α)/tg (π + α) · ctg (π/2 + α)/tg (α − 3π/2) = −sin α.
1) sin (π/4 + α) − cos (π/4 − α) = 0;
2) cos (π/6 − α) − sin (π/3 + α) = 0;
3) sin (3π/2 − α)/tg (π + α) · ctg (π/2 + α)/tg (α − 3π/2) = −sin α.
1) По формуле приведения cos (π/4 − α) = sin (π/2 − (π/4 − α)) = sin (π/4 + α). Следовательно, разность равна 0.
2) cos (π/6 − α) = sin (π/2 − (π/6 − α)) = sin (π/3 + α), поэтому разность равна 0.
3) sin (3π/2 − α) = −cos α; tg (π + α) = tg α; ctg (π/2 + α) = −tg α; tg (α − 3π/2) = tg (α − 3π/2 + 2π) = tg (α + π/2) = −ctg α. Левая часть равна (−cos α/tg α) · (−tg α/(−ctg α)) = (−cos α/tg α) · (tg α/ctg α) = −cos α/ctg α = −cos α · sin α/cos α = −sin α. Тождество доказано.
2) cos (π/6 − α) = sin (π/2 − (π/6 − α)) = sin (π/3 + α), поэтому разность равна 0.
3) sin (3π/2 − α) = −cos α; tg (π + α) = tg α; ctg (π/2 + α) = −tg α; tg (α − 3π/2) = tg (α − 3π/2 + 2π) = tg (α + π/2) = −ctg α. Левая часть равна (−cos α/tg α) · (−tg α/(−ctg α)) = (−cos α/tg α) · (tg α/ctg α) = −cos α/ctg α = −cos α · sin α/cos α = −sin α. Тождество доказано.