Step By Step Calculus » 5.3 - Other Trigonometric Functions

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Synopsis
Definition of Other Trigonometric Functions: Trigonometric functions describe the interplay between the unit circle, the concept of angle, and the coordinates of points on the circle. In particular, if you rotate a radius of the unit circle by an angle \theta \theta, starting from the positive x x-axis, so its endpoint arrives at the point P = (x, y) P = (x, y) on the unit circle, then
  • \tan \theta \tan \theta is the ratio of \dfrac{y}{x} \dfrac{y}{x}. It is an odd function of \theta \theta.
  • \sec \theta=\dfrac{1}{\cos \theta} \sec \theta=\dfrac{1}{\cos \theta}, \csc \theta=\dfrac{1}{\sin \theta} \csc \theta=\dfrac{1}{\sin \theta}, and \cot \theta=\dfrac{1}{\tan \theta} \cot \theta=\dfrac{1}{\tan \theta}.
Standard Values: The values of tangent, cotangent, secant and cosecant for the standard angles are
\[ \begin{array}{c|c|c|c|c|c} \theta & 0 & \frac{\pi}{6} & \frac{\pi}{4} & \frac{\pi}{3} & \frac{\pi}{2} \\\hline \sec(\theta) & 1 & \frac{2\sqrt3}{3} & \sqrt2 & 2 & UDE \\\hline \csc(\theta) & UDE & 2 & \sqrt2 & \frac{2\sqrt3}{3} & 1 \\\hline \tan(\theta) & 0 & \frac{\sqrt3}{3} & 1 & \sqrt3 & UDE \\\hline \cot(\theta) & UDE & \sqrt3 & 1 & \frac{\sqrt3}{3} & 0 \\\hline \end{array} \]
Signs: The signs of tangent, cotangent, secant and cosecant are described by the following picture:

Periodicity: The periodicity of tangent, cotangent, secant and cosecant follows from the periodicity of cosine and sine, but is not always of period 2\pi2\pi:
(Periodicity of Sec) \sec \left(\theta +2\pi \right)=\sec \left(\theta x\right) \sec \left(\theta +2\pi \right)=\sec \left(\theta x\right)
(Periodicity of Csc) \csc \left(\theta +2\pi \right)=\csc \left(\theta \right) \csc \left(\theta +2\pi \right)=\csc \left(\theta \right)
(Periodicity of Tan) \tan \left(\theta +\pi \right)=\tan \left(\theta \right) \tan \left(\theta +\pi \right)=\tan \left(\theta \right)
(Periodicity of Cot) \cot \left(\theta +\pi \right)=\cot \left(\theta \right) \cot \left(\theta +\pi \right)=\cot \left(\theta \right)
Symmetry: The symmetry properties of tangent, cotangent, secant and cosecant are as follows:
(Evenness of Sec) \sec \left(-\theta \right)=\sec \left(\theta \right) \sec \left(-\theta \right)=\sec \left(\theta \right)
(Oddness of Csc) \csc \left(-\theta \right)=-\csc \left(\theta \right) \csc \left(-\theta \right)=-\csc \left(\theta \right)
(Oddness of Tan) \tan \left(-\theta \right)=-\tan \left(\theta \right) \tan \left(-\theta \right)=-\tan \left(\theta \right)
(Oddness of Cot) \cot \left(-\theta \right)=-\cot \left(\theta \right) \cot \left(-\theta \right)=-\cot \left(\theta \right)
By reflecting in the yy-axis, we can show that
( y y-axis reflection of Sec) \sec \left(\pi -\theta \right)=-\sec \left(\theta \right) \sec \left(\pi -\theta \right)=-\sec \left(\theta \right)
( y y-axis reflection of Csc) \csc \left(\pi -\theta \right)=\csc \left(\theta \right) \csc \left(\pi -\theta \right)=\csc \left(\theta \right)
( y y-axis reflection of Tan) \tan \left(\pi -\theta \right)=-\tan \left(\theta \right) \tan \left(\pi -\theta \right)=-\tan \left(\theta \right)
( y y-axis reflection of Cot) \cot \left(\pi -\theta \right)=-\cot \left(\theta \right) \cot \left(\pi -\theta \right)=-\cot \left(\theta \right)
Hyperbolic Identity: By dividing the circle identity by \cos^2(t)\cos^2(t) and \sin^2(t)\sin^2(t) respectively and rearranging, we get the hyperbolic identities:
(Tan/Sec Hyperbolic Identity) \sec^{2} \left(t \right)-\tan^{2} \left(t \right) \sec^{2} \left(t \right)-\tan^{2} \left(t \right) = = 1 1
(Cot/Csc Hyperbolic Identity) \csc^{2} \left(t \right)-\cot^{2} \left(t \right) \csc^{2} \left(t \right)-\cot^{2} \left(t \right) = = 1 1