Справочник. Тригонометрические формулы

Тригонометрические формулы

содержание справочника

Основные тригонометрические тождества

  1. \(\sin^2 x+\cos^2 x=1\)
  2. \(\mathrm{tg} x\cdot \mathrm{ctg} x=1\)
  3. \(\mathrm{tg} x=\displaystyle\frac{\sin x}{\cos x}\)
  4. \(\mathrm{ctg} x=\displaystyle\frac{\cos x}{\sin x}\)
  5. \(1+\mathrm{tg}^2 x=\displaystyle\frac{1}{\cos^2 x}\)
  6. \(1+\mathrm{ctg}^2 x=\displaystyle\frac{1}{\sin^2 x}\)

Формулы сложения

  1. \(\sin (x+y)=\sin x\cos y+\cos x\sin y\)
  2. \(\sin (x-y)=\sin x\cos y-\cos x\sin y\)
  3. \(\cos (x+y)=\cos x\cos y-\sin x\sin y\)
  4. \(\cos (x-y)=\cos x\cos y+\sin x\sin y\)
  5. \(\mathrm{tg} (x+y)=\displaystyle\frac{\mathrm{tg} x+\mathrm{tg} y}{1-\mathrm{tg} x \cdot\mathrm{tg} y}\)
  6. \(\mathrm{tg} (x-y)=\displaystyle\frac{\mathrm{tg} x-\mathrm{tg} y}{1+\mathrm{tg} x\cdot \mathrm{tg} y}\)
  7. \(\mathrm{ctg} (x+y)=\displaystyle\frac{\mathrm{ctg} x\cdot \mathrm{ctg} y-1}{\mathrm{ctg} y +\mathrm{ctg} x}\)
  8. \(\mathrm{ctg} (x-y)=\displaystyle\frac{\mathrm{ctg} x\cdot \mathrm{ctg} y+1}{\mathrm{ctg} y-\mathrm{ctg} x}\)

Формулы тригонометрических функций двойного аргумента

  1. \(\sin 2x=2\sin x\cos x\)
  2. \(\cos 2x=\cos^2 x-\sin^2 x\)
  3. \(\cos 2x=2\cos^2 x-1\)
  4. \(\cos 2x=1-2\sin^2 x\)
  5. \(\sin 2x=\displaystyle\frac{2\mathrm{tg} x}{1+\mathrm{tg}^2 x}\)
  6. \(\cos 2x=\displaystyle\frac{1-\mathrm{tg}^2 x}{1+\mathrm{tg}^2 x}\)
  7. \(\mathrm{tg} 2x=\displaystyle\frac{2\mathrm{tg} x}{1-\mathrm{tg}^2 x}\)
  8. \(\mathrm{ctg} 2x=\displaystyle\frac{\mathrm{ctg}^2 x-1}{2\mathrm{ctg} x}\)
  9. \((\sin x\pm\cos x)^2=1\pm\sin 2x\)
  10. \(\mathrm{tg} x=\displaystyle\frac{\sin 2x}{1+\cos 2x}=\frac{1-\cos 2x}{\sin 2x}\)

Формулы понижения степени

  1. \(\sin^2 x=\displaystyle\frac{1-\cos 2x}{2}\)
  2. \(\cos^2 x=\displaystyle\frac{1+\cos 2x}{2}\)
  3. \(\mathrm{tg}^2 x=\displaystyle\frac{1-\cos 2x}{1+\cos 2x}\)
  4. \(\sin^4x=\displaystyle\frac{1}{8}(3-4\cos 2x+\cos 4x)\)
  5. \(\cos^4x=\displaystyle\frac{1}{8}(3+4\cos 2x+\cos 4x)\)

Формулы тройного аргумента

  1. \(\sin 3x=3\sin x-4\sin^3 x\)
  2. \(\cos 3x=4\cos^3 x-3\cos x\)
  3. \(\mathrm{tg} 3x=\displaystyle\frac{3\mathrm{tg} x-\mathrm{tg}^3 x}{1-3\mathrm{tg}^2 x}\)

Преобразование сумм и разностей

  1. \(\sin x+\sin y=2\sin\frac{x+y}{2}\cos\frac{x-y}{2}\)
  2. \(\sin x-\sin y=2\sin\frac{x-y}{2}\cos\frac{x+y}{2}\)
  3. \(\cos x+\cos y=2\cos\frac{x+y}{2}\cos\frac{x-y}{2}\)
  4. \(\cos x-\cos y=-2\sin\frac{x+y}{2}\sin\frac{x-y}{2}\)
  5. \(\mathrm{tg} x+\mathrm{tg} y=\displaystyle\frac{\sin (x+y)}{\cos x\cos y}\)
  6. \(\mathrm{tg} x-\mathrm{tg} y=\displaystyle\frac{\sin (x-y)}{\cos x\cos y}\)
  7. \(\mathrm{ctg} x+\mathrm{ctg} y=\displaystyle\frac{\sin (x+y)}{\sin x\sin y}\)
  8. \(\mathrm{ctg} x-\mathrm{ctg} y=-\displaystyle\frac{\sin (x-y)}{\sin x\sin y}\)
  9. \(a\sin x+b\cos x=\sqrt{a^2+b^2}\sin (x+\alpha)\), где \(\alpha\) - угол, для которого \(\cos\alpha=\displaystyle\frac{a}{\sqrt{a^2+b^2}}\) и \(\sin\alpha=\displaystyle\frac{b}{\sqrt{a^2+b^2}}\)

Преобразование произведения

  1. \(\sin x\sin y=\displaystyle\frac{1}{2}(\cos (x-y)-\cos (x+y))\)
  2. \(\sin x\cos y=\displaystyle\frac{1}{2}(\sin (x-y)+\sin (x+y))\)
  3. \(\cos x\cos y=\displaystyle\frac{1}{2}(\cos (x-y)+\cos (x+y))\)

Секанс и косеканс

  1. \(\sec{x}=\displaystyle\frac{1}{\cos{x}}\)
  2. \(\mathrm{cosec}x=\displaystyle\frac{1}{\sin{x}}\)