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Precalculus: Trigonometric Identities and Sum/Difference Formulas

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Terms in this set (21)

  • What is the Pythagorean identity involving sine and cosine?
    The identity is \(\sin^2\theta + \cos^2\theta = 1\).
  • State the identity for tangent in terms of sine and cosine.
    Tangent is defined as \(\tan\theta = \frac{\sin\theta}{\cos\theta}\).
  • What is the sum formula for sine: sin(a + b)?
    \(\sin(a+b) = \sin a \cos b + \cos a \sin b\).
  • What is the difference formula for sine: sin(a - b)?
    \(\sin(a-b) = \sin a \cos b - \cos a \sin b\).
  • What is the sum formula for cosine: cos(a + b)?
    \(\cos(a+b) = \cos a \cos b - \sin a \sin b\).
  • What is the difference formula for cosine: cos(a - b)?
    \(\cos(a-b) = \cos a \cos b + \sin a \sin b\).
  • State the sum formula for tangent: tan(a + b).
    \(\tan(a+b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\).
  • State the difference formula for tangent: tan(a - b).
    \(\tan(a-b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}\).
  • What is the double-angle formula for sine: sin(2θ)?
    \(\sin(2\theta) = 2 \sin\theta \cos\theta\).
  • What is the double-angle formula for cosine: cos(2θ)?
    There are three forms: \(\cos(2\theta) = \cos^2\theta - \sin^2\theta\), \(= 2\cos^2\theta - 1\), and \(= 1 - 2\sin^2\theta\).
  • What is the double-angle formula for tangent: tan(2θ)?
    \(\tan(2\theta) = \frac{2 \tan\theta}{1 - \tan^2\theta}\).
  • State the reciprocal identity for sine.
    Cosecant is the reciprocal of sine: \(\csc\theta = \frac{1}{\sin\theta}\).
  • State the reciprocal identity for cosine.
    Secant is the reciprocal of cosine: \(\sec\theta = \frac{1}{\cos\theta}\).
  • State the reciprocal identity for tangent.
    Cotangent is the reciprocal of tangent: \(\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}\).
  • What is the co-function identity for sine and cosine?
    \(\sin(90^\circ - \theta) = \cos\theta\) and \(\cos(90^\circ - \theta) = \sin\theta\).
  • What is the co-function identity for tangent and cotangent?
    \(\tan(90^\circ - \theta) = \cot\theta\) and \(\cot(90^\circ - \theta) = \tan\theta\).
  • What is the identity for sine of a negative angle?
    \(\sin(-\theta) = -\sin\theta\).
  • What is the identity for cosine of a negative angle?
    \(\cos(-\theta) = \cos\theta\).
  • What is the identity for tangent of a negative angle?
    \(\tan(-\theta) = -\tan\theta\).
  • What is the product-to-sum formula for sin a sin b?
    \(\sin a \sin b = \frac{1}{2}[\cos(a-b) - \cos(a+b)]\).
  • What is the product-to-sum formula for cos a cos b?
    \(\cos a \cos b = \frac{1}{2}[\cos(a-b) + \cos(a+b)]\).