Глава V. § 31. Формулы приведения — с. 159
524. Найти значение острого угла α, если:
1) cos 75° = cos (90° − α);
2) sin 150° = sin (90° + α);
3) sin 150° = sin (180° − α);
4) cos 310° = cos (270° + α);
5) sin 5/4 π = sin (π + α);
6) tg π/5 = tg (π/2 − α);
7) cos 7π/4 = cos (3/2 π + α);
8) ctg 11/6 π = ctg (2π − α).
1) cos 75° = cos (90° − α);
2) sin 150° = sin (90° + α);
3) sin 150° = sin (180° − α);
4) cos 310° = cos (270° + α);
5) sin 5/4 π = sin (π + α);
6) tg π/5 = tg (π/2 − α);
7) cos 7π/4 = cos (3/2 π + α);
8) ctg 11/6 π = ctg (2π − α).
1) cos (90° − α) = sin α, значит sin α = cos 75° = sin 15°, α = 15°.
2) sin (90° + α) = cos α, sin 150° = sin 30° = 1/2, cos α = 1/2, α = 60°.
3) sin (180° − α) = sin α, поэтому sin α = sin 150° = sin 30° = 1/2, α = 30°.
4) cos (270° + α) = sin α; cos 310° = cos 50° = sin 40°. Тогда sin α = sin 40°, α = 40°.
5) sin (π + α) = −sin α; sin 5π/4 = −sin π/4 = −√2/2. Тогда sin α = √2/2, α = π/4.
6) tg (π/2 − α) = ctg α, ctg α = tg π/5 = ctg (π/2 − π/5) = ctg 3π/10, α = 3π/10.
7) cos (3π/2 + α) = sin α; cos 7π/4 = cos π/4 = √2/2. Тогда sin α = √2/2, α = π/4.
8) ctg (2π − α) = −ctg α; ctg 11π/6 = ctg (2π − π/6) = −ctg π/6 = −√3. Тогда ctg α = √3, α = π/6.
Ответ: 1) 15°; 2) 60°; 3) 30°; 4) 40°; 5) π/4; 6) 3π/10; 7) π/4; 8) π/6.
2) sin (90° + α) = cos α, sin 150° = sin 30° = 1/2, cos α = 1/2, α = 60°.
3) sin (180° − α) = sin α, поэтому sin α = sin 150° = sin 30° = 1/2, α = 30°.
4) cos (270° + α) = sin α; cos 310° = cos 50° = sin 40°. Тогда sin α = sin 40°, α = 40°.
5) sin (π + α) = −sin α; sin 5π/4 = −sin π/4 = −√2/2. Тогда sin α = √2/2, α = π/4.
6) tg (π/2 − α) = ctg α, ctg α = tg π/5 = ctg (π/2 − π/5) = ctg 3π/10, α = 3π/10.
7) cos (3π/2 + α) = sin α; cos 7π/4 = cos π/4 = √2/2. Тогда sin α = √2/2, α = π/4.
8) ctg (2π − α) = −ctg α; ctg 11π/6 = ctg (2π − π/6) = −ctg π/6 = −√3. Тогда ctg α = √3, α = π/6.
Ответ: 1) 15°; 2) 60°; 3) 30°; 4) 40°; 5) π/4; 6) 3π/10; 7) π/4; 8) π/6.
525. Вычислить с помощью формулы приведения:
1) cos 150°;
2) sin 135°;
3) ctg 135°;
4) cos 120°;
5) cos 225°;
6) sin 210°;
7) ctg 240°;
8) sin 315°.
1) cos 150°;
2) sin 135°;
3) ctg 135°;
4) cos 120°;
5) cos 225°;
6) sin 210°;
7) ctg 240°;
8) sin 315°.
1) cos 150° = cos (180° − 30°) = −cos 30° = −√3/2.
2) sin 135° = sin (180° − 45°) = sin 45° = √2/2.
3) ctg 135° = ctg (180° − 45°) = −ctg 45° = −1.
4) cos 120° = cos (180° − 60°) = −cos 60° = −1/2.
5) cos 225° = cos (180° + 45°) = −cos 45° = −√2/2.
6) sin 210° = sin (180° + 30°) = −sin 30° = −1/2.
7) ctg 240° = ctg (180° + 60°) = ctg 60° = √3/3.
8) sin 315° = sin (360° − 45°) = −sin 45° = −√2/2.
2) sin 135° = sin (180° − 45°) = sin 45° = √2/2.
3) ctg 135° = ctg (180° − 45°) = −ctg 45° = −1.
4) cos 120° = cos (180° − 60°) = −cos 60° = −1/2.
5) cos 225° = cos (180° + 45°) = −cos 45° = −√2/2.
6) sin 210° = sin (180° + 30°) = −sin 30° = −1/2.
7) ctg 240° = ctg (180° + 60°) = ctg 60° = √3/3.
8) sin 315° = sin (360° − 45°) = −sin 45° = −√2/2.
526. Вычислить с помощью формулы приведения:
1) tg 5π/4;
2) sin 7π/6;
3) cos 5π/3;
4) ctg 5π/3;
5) sin (−13π/6);
6) cos (−7π/3);
7) tg (−2π/3);
8) ctg (−7π/4).
1) tg 5π/4;
2) sin 7π/6;
3) cos 5π/3;
4) ctg 5π/3;
5) sin (−13π/6);
6) cos (−7π/3);
7) tg (−2π/3);
8) ctg (−7π/4).
1) tg 5π/4 = tg (π + π/4) = tg π/4 = 1.
2) sin 7π/6 = sin (π + π/6) = −sin π/6 = −1/2.
3) cos 5π/3 = cos (2π − π/3) = cos π/3 = 1/2.
4) ctg 5π/3 = ctg (2π − π/3) = −ctg π/3 = −√3/3.
5) sin (−13π/6) = −sin 13π/6 = −sin (2π + π/6) = −sin π/6 = −1/2.
6) cos (−7π/3) = cos 7π/3 = cos (2π + π/3) = cos π/3 = 1/2.
7) tg (−2π/3) = −tg 2π/3 = −tg (π − π/3) = tg π/3 = √3.
8) ctg (−7π/4) = −ctg 7π/4 = −ctg (2π − π/4) = ctg π/4 = 1.
Ответ: 1) 1; 2) −1/2; 3) 1/2; 4) −√3/3; 5) −1/2; 6) 1/2; 7) √3; 8) 1.
2) sin 7π/6 = sin (π + π/6) = −sin π/6 = −1/2.
3) cos 5π/3 = cos (2π − π/3) = cos π/3 = 1/2.
4) ctg 5π/3 = ctg (2π − π/3) = −ctg π/3 = −√3/3.
5) sin (−13π/6) = −sin 13π/6 = −sin (2π + π/6) = −sin π/6 = −1/2.
6) cos (−7π/3) = cos 7π/3 = cos (2π + π/3) = cos π/3 = 1/2.
7) tg (−2π/3) = −tg 2π/3 = −tg (π − π/3) = tg π/3 = √3.
8) ctg (−7π/4) = −ctg 7π/4 = −ctg (2π − π/4) = ctg π/4 = 1.
Ответ: 1) 1; 2) −1/2; 3) 1/2; 4) −√3/3; 5) −1/2; 6) 1/2; 7) √3; 8) 1.