Question

How many of following statements are true?

(1)Period of sin $\theta$ is $\pi$, because sin 0 and sin $\pi$ = 0

(2) Period of cos $\theta$ is $\pi$

(3) Period of tan $\theta$ is $\pi$

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Answer 1

Solution: A function f(x) is said to be periodic with period P if f(x+p) = f(x) for all x in the domain of f(x). The last part, for all x the domain is important.
So even if sin (0+$\pi$) = sin 0 = 0, $\pi$ can't be the period of sin x, because
Sin(x+$\pi$) = -sin x $\ne$ sin x. If we take $\frac{\pi }{2}$, instead of 0, we get sin ($\pi$+ $\frac{\pi }{2}$) = sin $\frac{3\pi }{2}$ = -1. This is not equal to sin $\frac{\pi }{2}$
The period of sinx and cosx is 2$\pi$, because sin (x + 2$\pi$) = sinx and cos (x + 2$\pi$) = cosx. 2$\pi$ is the fundamental period of these functions. (Fundamental period is the least value of P (>0) for which the function repeats).
Similarly,
Period of tanx is n because tan ($\pi$ + x) = tan x