Formulas and Tables

Trigonometry

Basic Trigonometric Identities

Angle: \(\alpha\)
Trigonometric functions: \(\sin \alpha,\) \(\cos \alpha,\) \(\tan \alpha,\) \(\cot \alpha,\) \(\sec \alpha,\) \(\csc \alpha\)

Set of integers: \(\mathbb{Z}\)
Integers: \(n\)

  1. Trigonometric identities establish a connection between trigonometric functions of the same argument (angle \(\alpha\)).
  2. Pythagorean trigonometric identity
    \({\sin ^2}\alpha + {\cos ^2}\alpha = 1\)
    This identity is the result of application of the Pythagorean theorem to a triangle in the unit circle.
  3. Relationship between the cosine and tangent
    \(\large{\frac{1}{{\cos^2}\alpha}}\normalsize – {\tan ^2}\alpha = 1\) or \(\sec^2\alpha – {\tan ^2}\alpha = 1.\)
    This identity follows from the Pythagorean trigonometric identity and is obtained by dividing the left and right sides by \(\cos^2 \alpha\). It is assumed that \(\alpha \ne \large{\frac{\pi}{2}}\normalsize + \pi n,\) \(n \in \mathbb{Z}\).
  4. Relationship between the sine and cotangent
    \(\large{\frac{1}{{\sin^2}\alpha}}\normalsize – {\cot ^2}\alpha = 1\) or \(\csc^2\alpha – {\cot ^2}\alpha = 1.\)
    This formula also follows from the Pythagorean trigonometric identity (it is obtained by dividing the left and right sides by \(\sin^2 \alpha\). It is assumed that \(\alpha \ne \pi n,\) \(n \in \mathbb{Z}\).
  5. Definition of tangent
    \(\tan \alpha = \large{\frac{\sin \alpha}{\cos \alpha}}\normalsize ,\) where \(\alpha \ne \large{\frac{\pi}{2}}\normalsize + \pi n,\) \(n \in \mathbb{Z}\).
  6. Definition of cotangent
    \(\cot \alpha = \large{\frac{\cos \alpha}{\sin \alpha}}\normalsize ,\) where \(\alpha \ne \pi n,\) \(n \in \mathbb{Z}\).
  7. Consequence of the definitions of tangent and cotangent
    \(\tan \alpha \cdot \cot \alpha = 1,\) where \(\alpha \ne \large{\frac{\pi n}{2}}\normalsize ,\) \(n \in \mathbb{Z}.\)
  8. Definition of secant
    \(\sec \alpha = \large{\frac{1}{\cos \alpha}}\normalsize,\) \(\alpha \ne \large{\frac{\pi}{2}}\normalsize +\pi n,\) \(n \in \mathbb{Z}.\)
  9. Definition of cosecant
    \(\csc \alpha = \large{\frac{1}{\sin \alpha}}\normalsize,\) \(\alpha \ne \pi n,\) \(n \in \mathbb{Z}.\)