Precalculus Review

Sine and cosine function

The function $y = \sin x$ and $y = \cos x$ are respectively called sine and cosine function. The domain of both functions is ${D = (-\infty, \infty)}$ , and their range is

. But the sine function is odd: $\sin(-x) = - \sin x$ , and the cosine function is even: $\cos(-x) = \cos x$ .

(S) $y = \sin x$

$\includegraphics{lec10a.ps}$ $\includegraphics{lec10b.ps}$

(C) $y = \cos x$

$\includegraphics{lec10c.ps}$

Periodic function. A function is said to be periodic if it satisfies for every with some constant value . The least such positive value for which holds is called the period of . The sine function and the cosine function are periodic, and both have the period of $2 \pi$ : $\sin(x + 2\pi) = \sin x$ and $\cos(x + 2\pi) = \cos x$ . Moreover, we can observe that $\sin(x + 2\pi n) = \sin x$ and $\cos(x + 2\pi n) = \cos x$ for any integer . The immediate application is to find the coordinate $(\cos\theta, \sin\theta) = (\cos\theta_R, \sin\theta_R)$ on the unit circle by using the “reference angle $\theta_R$ ” (the acute angle between the terminal side of the original angle $\theta$ and the -axis). The value $\pi$ is not the period for the sine and the cosine functions, but rather it gives the phase shift: $\sin(x + \pi) = -\sin x$ and $\cos(x + \pi) = -\cos x$ .

	(S) $y = \sin x$
$\includegraphics{lec10a.ps}$	$\includegraphics{lec10b.ps}$
	(C) $y = \cos x$
	$\includegraphics{lec10c.ps}$