Define a period and the least positive period, including the requirement that the domain is unchanged by translation. Use unit-circle coordinates to construct both graphs on \([0,2\pi]\). Give their zeros, extrema and parity; list the least periods of all six trigonometric functions. Complete the quarter-cycle table and evaluate \(\sin(\dfrac{37 \pi}{6}),\ \cos(\dfrac{37 \pi}{6})\). Explain separately why the constant function \(f(x)=3\) has no least positive period. Curve 1: \(\sin{\left(x \right)}\). Curve 2: \(\cos{\left(x \right)}\). Show the requested calculations and reasoning. Written explanations and sketches require teacher review; choosing a summary does not verify that work. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
Answer: Sine/cosine least period \(2 \pi\); parities odd/even; translated values \(\dfrac{1}{2}\) and \(\dfrac{\sqrt{3}}{2}\).
A period \(p>0\) satisfies \(D+p=D\) and \(f(x+p)=f(x)\) for every \(x\in D\). The least positive such number, when it exists, is the fundamental period. One full circuit of the unit circle adds \(2\pi\) to the input without changing either coordinate. The first return of both sine value and orientation, or cosine value and orientation, occurs after \(2\pi\).
Input
Sine
Cosine
\(0\)
\(0\)
\(1\)
\(\dfrac{\pi}{2}\)
\(1\)
\(0\)
\(\pi\)
\(0\)
\(-1\)
\(\dfrac{3 \pi}{2}\)
\(-1\)
\(0\)
\(2 \pi\)
\(0\)
\(1\)
\(\sin x=0\iff x=n\pi\); sine maxima are at \(x=\dfrac{\pi}{2}+2n\pi\) and minima at \(x=\dfrac{3\pi}{2}+2n\pi\). \(\cos x=0\iff x=\dfrac{\pi}{2}+n\pi\); cosine maxima are at \(2n\pi\) and minima at \((2n+1)\pi\), for integers \(n\). Reflection of the terminal point gives \(\sin(-x)=-\sin x\) and \(\cos(-x)=\cos x\). Sine, cosine, secant and cosecant have least period \(2\pi\); tangent and cotangent have least period \(\pi\) on their pole-excluded domains. Subtract \(6 \pi\) from the translated angle to obtain \(\dfrac{\pi}{6}\). The values are \(\dfrac{1}{2}\) and \(\dfrac{\sqrt{3}}{2}\). The constant function on all real inputs has every positive number as a period. Given any proposed least period, half of it is a smaller positive period, so no least one exists. Written-work review:
Give the domain-preserving definition and distinguish a period from the least period.
Construct both full base cycles and the exact quarter-cycle table, including zeros, extrema and parity.
Reduce the translated angle and justify the constant-function exception.