Q.Find a cosine function whose period is 7.

A function f is called periodic if there is a positive constant p with $f(x+p)=f(x)$ for every x in its domain; the smallest such p is called the period. Trigonometric functions are periodic because rotating the angle θ by one full revolution (360° = 2π radians) returns the terminal ray — and hence the point P and every coordinate-based function value — to exactly where it started. This gives $\sin(\theta+2\pi)=\sin\theta$ and $\cos(\theta+2\pi)=\cos\theta$, so sine, cosine, and their reciprocals cosecant and secant, all have period $2\pi$. Tangent and cotangent behave differently: because $\tan(\theta+\pi)=\tan\theta$ (rotating by half a revolution sends P to its diametrically opposite point, which has the same ratio y/x since both coordinates flip sign), their fundamental period is th …