Step By Step Calculus » 5.2 - Sine and Cosine Functions

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Synopsis
Definition of Sine and Cosine 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 the angle \theta \theta from the positive x x-axis, so its endpoint arrives at the point P = (x, y) P = (x, y) on the unit circle, then the cosine and sine functions are just the x x and y y coordinates respectively of P:
  • \sin \theta \sin \theta is the y y co-ordinate. It is an odd function of \theta \theta.
  • \cos \theta \cos \theta is the x x co-ordinate. It is an even function of \theta \theta.
Periodicity Both the sine and cosine function are periodic with the period 2\pi2\pi i.e.
(Periodicity of Cos) \cos \left(\theta +2\pi \right)=\cos \left(\theta \right) \cos \left(\theta +2\pi \right)=\cos \left(\theta \right)
(Periodicity of Sin) \sin \left(x+2\pi \right)=\sin \left(x\right) \sin \left(x+2\pi \right)=\sin \left(x\right)
Symmetry The points (\cos(\theta),\sin(\theta))(\cos(\theta),\sin(\theta)) and (\cos(-\theta),\sin(-\theta))(\cos(-\theta),\sin(-\theta)) on the unit circle are reflections of each other across the xx-axis. This gives us the properties of evenness of cosine and oddness of sine.
(Evenness of Cos) \cos \left(-\theta \right)=\cos \left(\theta \right) \cos \left(-\theta \right)=\cos \left(\theta \right)
(Oddness of Sin) \sin \left(-\theta \right)=-\sin \left(\theta \right) \sin \left(-\theta \right)=-\sin \left(\theta \right).
The reflection of the point (\cos(\theta),\sin(\theta))(\cos(\theta),\sin(\theta)) across the yy-axis gives us
( y y-axis reflection of Cos) \cos \left(\pi -\theta \right)=-\cos \left(\theta \right) \cos \left(\pi -\theta \right)=-\cos \left(\theta \right)
( y y-axis reflection of Sin) \sin \left(\pi -\theta \right)=\sin \left(\theta \right) \sin \left(\pi -\theta \right)=\sin \left(\theta \right).
The successive reflection of the point (\cos(\theta),\sin(\theta))(\cos(\theta),\sin(\theta)) across the xx and yy axes (in either order) gives us
(Opposing Point) \cos \left(\pi +\theta \right)=-\cos \left(\theta \right) \cos \left(\pi +\theta \right)=-\cos \left(\theta \right)
(Opposing Point) \sin \left(\pi +\theta \right)=-\sin \left(\theta \right) \sin \left(\pi +\theta \right)=-\sin \left(\theta \right).
Reference Angles
For angles \theta\theta in quadrants II, III or IV, there is a reference angle \theta_R\theta_R in quadrant I. The sine and cosine of the reference angle are related to the sine and cosine of the angle \theta\theta as follows:
\cos \left(\theta \right)=-\cos \left(\theta ^{R}\right) \cos \left(\theta \right)=-\cos \left(\theta ^{R}\right) \sin \left(\theta \right)=\sin \left(\theta ^{R}\right) \sin \left(\theta \right)=\sin \left(\theta ^{R}\right) \theta \theta in Quadrant II
\cos \left(\theta \right)=-\cos \left(\theta ^{R}\right) \cos \left(\theta \right)=-\cos \left(\theta ^{R}\right) \sin \left(\theta \right)=-\sin \left(\theta ^{R}\right) \sin \left(\theta \right)=-\sin \left(\theta ^{R}\right) \theta \theta in Quadrant III
\cos \left(\theta \right)=\cos \left(\theta ^{R}\right) \cos \left(\theta \right)=\cos \left(\theta ^{R}\right) \sin \left(\theta \right)=-\sin \left(\theta ^{R}\right) \sin \left(\theta \right)=-\sin \left(\theta ^{R}\right) \theta \theta in Quadrant IV.
Sin/Cos Circle Identity Since every point on the unit circle centered at the origin can be expressed by (x,y) = (\cos(t),\sin(t))(x,y) = (\cos(t),\sin(t)) we have
\displaystyle \textbf{(Sin/Cos Circle Identity) }\ \sin^{2} \left(t \right)+\cos^{2} \left(t \right)=1.
\displaystyle \textbf{(Sin/Cos Circle Identity) }\ \sin^{2} \left(t \right)+\cos^{2} \left(t \right)=1.
This can also be termed as Pythagorean identity, another instance of Pythagorean theorem.
Cos and Sin for the Standard Angles The following table shows the exact value of sine and cosine for standard angles.
\newcommand{\T}{\rule{0pt}{2.7ex}} \newcommand{\B}{\rule[-1.7ex]{0pt}{0pt}} \[ \begin{array}{|c|c|c|c|c|c|}\hline \T\B\theta & 0 & \frac{\pi}{6} & \frac{\pi}{4}& \frac{\pi}{3} & \frac{\pi}{2} \\\hline \T\B\cos(\theta) & 1 & \frac{\sqrt3}{2} & \frac{\sqrt2}{2} & \frac12 & 0 \\\hline \T\B\sin(\theta) & 0 & \frac12 & \frac{\sqrt2}{2} & \frac{\sqrt3}{2} & 1 \\\hline \end{array} \]