Formulas

Trig Identities

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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
      \(\csc \alpha = \large{\frac{1}{\sin \alpha}}\normalsize,\) \(\alpha \ne \pi n,\) \(n \in \mathbb{Z}.\)