Question

copy and complete the table of properties for ( y = sin x ) and ( y = cos x ) for all real numbers.

property	( y = sin x )	( y = cos x )
minimum
amplitude
period
domain
range
( y )-intercept
( x )-intercepts

Explanation:

Step1: Analyze the maximum value

For \(y = \sin x\), the maximum value occurs when \(x=\frac{\pi}{2}+ 2k\pi,k\in\mathbb{Z}\), and \(y = 1\). For \(y=\cos x\), the maximum value occurs when \(x = 2k\pi,k\in\mathbb{Z}\), and \(y = 1\).

Step2: Analyze the minimum value

For \(y=\sin x\), the minimum value occurs when \(x=\frac{3\pi}{2}+2k\pi,k\in\mathbb{Z}\), and \(y=- 1\). For \(y=\cos x\), the minimum value occurs when \(x=(2k + 1)\pi,k\in\mathbb{Z}\), and \(y=-1\).

Step3: Analyze the amplitude

The amplitude of \(y = A\sin x\) and \(y=A\cos x\) is \(|A|\). For \(y=\sin x\) (\(A = 1\)) and \(y=\cos x\) (\(A = 1\)), the amplitude is \(\frac{1-(-1)}{2}=1\).

Step4: Analyze the period

The period of \(y=\sin(Bx)\) and \(y=\cos(Bx)\) is \(T=\frac{2\pi}{|B|}\). For \(y=\sin x\) (\(B = 1\)) and \(y=\cos x\) (\(B = 1\)), the period \(T = 2\pi\).

Step5: Analyze the domain

Since \(y=\sin x\) and \(y=\cos x\) are defined for all real - valued \(x\), the domain is \(x\in(-\infty,\infty)\) or \(\mathbb{R}\).

Step6: Analyze the range

Since the minimum value is \(-1\) and the maximum value is \(1\), the range is \(y\in[-1,1]\).

Step7: Analyze the \(y\) - intercept

For \(y=\sin x\), when \(x = 0\), \(y=\sin(0)=0\). For \(y=\cos x\), when \(x = 0\), \(y=\cos(0)=1\).

Step8: Analyze the \(x\) - intercepts

For \(y=\sin x\), \(\sin x=0\) when \(x = k\pi,k\in\mathbb{Z}\). For \(y=\cos x\), \(\cos x=0\) when \(x=\frac{\pi}{2}+k\pi,k\in\mathbb{Z}\).

Answer:

Property	\(y = \sin x\)	\(y=\cos x\)
minimum	\(-1\)	\(-1\)
amplitude	\(1\)	\(1\)
period	\(2\pi\)	\(2\pi\)
domain	\(\mathbb{R}\)	\(\mathbb{R}\)
range	\([-1,1]\)	\([-1,1]\)
\(y\) - intercept	\(0\)	\(1\)
\(x\) - intercepts	\(x = k\pi,k\in\mathbb{Z}\)	\(x=\frac{\pi}{2}+k\pi,k\in\mathbb{Z}\)