Trigonometry

Table of Contents

1. Trigonometric Functions

1.1. atan2

The angle between the positive \( x \)-axis and the ray from the origin to the point \( (x,y) \).

The function for the actual angle.

2. Pythagorean Identity

  • \[ \tan \theta + \cot \theta = \sec\theta\csc\theta \]

3. Law of Sines

  • \[ \frac{\sin \alpha}{a} = \frac{\sin \beta}{b} = \frac{\sin\gamma }{c} = \frac{1}{2R} \]
  • where \(R\) is the circumradius.

4. Law of Cosines

  • \[ c^2 = a^2 + b^2 -2ab\cos\gamma. \]

5. General Angle Sum Formula

  • For a series:
    • \[ \sum_{i=1}^\infty \theta_i \]
    • that converges abolutely,

5.1. Sine

  • \[ \sin\bigg(\sum_i\theta_i\bigg) = \sum_{\text{odd }k\ge 1}(-1)^{\frac{k-1}{2}} \sum_{A\subseteq \{1,2,3,\dots\}\atop |A| = k}\left( \prod_{i\in A} \sin\theta_i\prod_{i\not\in A}\cos\theta_i\right). \]

5.2. Cosine

  • \[ \cos\bigg(\sum_i\theta_i\bigg) = \sum_{\text{even }k\ge 0}(-1)^{\frac{k}{2}} \sum_{A\subseteq \{1,2,3,\dots\}\atop |A| = k}\left( \prod_{i\in A} \sin\theta_i\prod_{i\not\in A}\cos\theta_i\right). \]

5.3. Tangent

  • The ratio of the sine and cosine formula
  • \[ \tan\bigg(\sum_i\theta_i\bigg) = \frac{\displaystyle\sum_{\text{odd }k\ge 1}(-1)^{\frac{k-1}{2}} \sum_{A\subseteq \{1,2,3,\dots\}\atop |A| = k} \prod_{i\in A} \tan\theta_i}{\displaystyle\sum_{\text{even }k\ge 0}(-1)^{\frac{k}{2}} \sum_{A\subseteq \{1,2,3,\dots\}\atop |A| = k} \prod_{i\in A}\tan\theta_i}. \]
  • Using the elementary symmetric polynomials in \(\tan\theta_i\):
    • \[ \tan\bigg(\sum_i\theta_i\bigg) = \frac{e_1 - e_3 + e_5 -\cdots}{e_0 - e_2 + e_4 - \cdots} \]

5.4. Cotangent

  • The reciprocal of the tangent formula
  • In terms of \(\tan\theta_i\):
    • \[ \cot\bigg(\sum_i\theta_i\bigg) =\frac{e_0 - e_2 + e_4 - \cdots}{e_1 - e_3 + e_5 -\cdots} \]
    • It is not recommended to express it in terms of \(\cot\theta_i\), because for the infinite series \(\cot(\theta_i) \to \infty\). But for the finite case, it is possible using the fact:
      • \[ \left(\prod_{i=1}^n\cot\theta_i\right)e_k(\tan\theta_i) = e_{n-k}(\cot\theta_i) \]
      • since \(\tan\theta_i\cos\theta_i = 1\), except for singular points.
    • Threrefore, for odd \(n\), in terms of \(\cot\theta_i\):
      • \[ \cot\bigg(\sum_{i=1}^n\theta_i\bigg) = \frac{e_1 - e_3 + e_5 -\cdots + (-1)^{(n-1)/2}e_n}{e_0 - e_2 + e_4- \cdots +(-1)^{(n-1)/2} e_{n-1} } \]
    • And for even \(n\) it looks identical to the formula in \(\tan\theta_i\).

6. Sine Product Formula

  • First discovered by Leonhard Euler.
  • Obtained by multiplying all the binomials that has the same zero as the sine function
    • \[ \sin x=x\prod_{k=1}^\infty \left( 1-\frac{x^{2}}{\pi^{2}k^{2}} \right) . \]

7. Law of Cotangents

  • \[ \frac{\cot (\alpha /2)}{s-a} = \frac{\cot (\beta /2)}{s-b} = \frac{\cot (\gamma /2)}{s-c} = \frac{1}{r} \]
  • where \(s\) is the semiperimeter and \(r\) is the inradius which can be given by the inradius formula.

8. Mollweide's Formulae

  • They are readily proved using the addition formula of the sine and cosine, and the law of cotangents.

8.1. First

  • \[ \frac{a-b}{c} = \frac{\sin\frac{1}{2}(\alpha - \beta)}{\cos\frac{1}{2}\gamma} \]

8.2. Second

  • \[ \frac{b+a}{c} = \frac{\cos\frac{1}{2}(\alpha - \beta)}{\sin\frac{1}{2}\gamma} \]

9. Triple Tangent Identity

  • \[ \tan\alpha\tan\beta\tan\gamma = \tan\alpha+\tan\beta+\tan\gamma \]
    • when \(\alpha + \beta + \gamma = \pi\).
  • It is obtained via the general angle sum of tangent, using the fact that \(\tan(\pi) = 0\).

10. Triple Cotangent Identity

  • \[ \cot\frac{\alpha}{2}\cot\frac{\beta}{2}\cot\frac{\gamma}{2} = \cot\frac{\alpha}{2}+\cot\frac{\beta}{2}+\cot\frac{\gamma}{2} \]
    • when \(\alpha+\beta+\gamma = \pi\).
  • It is obtained via the general angle sum of cotangent, using the fact that \(\cot(\pi/2) = 0\).

11. Inverse Trigonometric Functions

11.1. Properties

  • It is only defined for the principal interval.

11.1.1. Composition

  • \(\sin(\cos^{-1} x) = \sqrt{1-x^2}\)
  • \(\sec(\tan^{-1}x) = \sqrt{1+x^2}\)

11.1.2. Derivatives

  • The denominator is the derivative of the normal function in terms of the normal function as \(x\).
  • \[ \frac{d}{dx}\sin^{-1}x = \frac{1}{\sqrt{1-x^2}} \]
  • \[ \frac{d}{dx}\tan^{-1}x = \frac{1}{1+x^2} \]
  • \[ \frac{d}{dx}\sec^{-1}x = \frac{1}{|x|\sqrt{x^2-1}} \]

12. Hyperbolic Functions

12.1. Definitions

  • \(\sinh x = (e^x-e^{-1})/2\)
  • \(\cosh x = (e^x + e^{-1})/2\)
  • \(\tanh x = \sinh x / \cosh x\)
  • \(\mathop{\rm sech} x = 1/\cosh x\)

12.2. Properties

  • \(\cosh^2 x - \sinh^2 x = 1\)
  • \(\tanh^2 x + \operatorname{sech}^2 x = 1\)
  • \(\sinh(x\pm y) = \sinh x\cosh y \pm \cosh x\sinh y\)
  • \(\cosh(x\pm y) = \cosh x\cosh y \pm \sinh x\sinh y\)
  • \[\tanh(x\pm y) = \frac{\tanh x\pm \tanh y}{1\pm \tanh x\tanh y}\]

12.2.1. Derivatives

  • \[ \frac{d}{dx}\sinh x = \cosh x \]
  • \[ \frac{d}{dx}\cosh x = \sinh x \]
  • \[ \frac{d}{dx}\tanh x = \operatorname{sech}^2 x \]
  • \[ \frac{d}{dx}\operatorname{sech} x = -\operatorname{sech} x\tanh x \]
  • \[ \frac{d}{dx}\operatorname{csch} x = -\operatorname{csch} x \operatorname{coth} x \]
  • \[ \frac{d}{dx} \operatorname{coth} x = - \operatorname{csch}^2 x \]

12.2.2. Antiderivatives

  • \[ \int \tanh x\, dx = \ln\left(e^x +e^{-x}\right) +C \]

13. Inverse Hyperbolic Functions

13.1. Properties

  • \(\sinh^{-1}x = \ln\left( x + \sqrt{x^2+1}\right)\)
  • \(\cosh^{-1}x = \ln\left(x+\sqrt{x^2-1}\right)\)

    • proof.

      \begin{align*} \cosh^{-1} &y = x \\ \implies &y+\sqrt{y^2 - 1} = e^x\\ \implies &y^2 - 1 = e^{2x} - 2ye^x + y^2 \\ \implies & y = \frac{e^{2x}+1}{2e^x} = \cosh x \end{align*}
  • \[ \tanh^{-1}x = \frac{1}{2}\ln\frac{1+x}{1-x} \]

13.1.1. Derivatives

  • \[ \frac{d}{dx}\sinh^{-1}x=\frac{1}{\sqrt{1+x^2}} \]
  • \[ \frac{d}{dx}\cosh^{-1}x = \frac{1}{\sqrt{x^2-1}} \]
  • \[ \frac{d}{dx}\tanh^{-1}x = \frac{1}{1-x^2} \]

14. References

Created: 2025-05-04 Sun 10:10