Question

1. consider the functions ( y=sin x ) and ( y=sin ^{-1} x ).

(a) fill in the table below for the sine function on the restricted interval ( left\frac{-pi}{2}, \frac{pi}{2}
ight ).

(b) using your table in part (a) construct a table for ( y=sin ^{-1} x ).

(c) plot and label the points from both tables on the axes below.

Explanation:

Step1: Calculate $\sin(x)$ values

• For \(x =-\frac{\pi}{2}\), \(\sin(-\frac{\pi}{2})=- 1\) (using the property \(\sin(-\alpha)=-\sin\alpha\) and \(\sin(\frac{\pi}{2}) = 1\))
• For \(x = 0\), \(\sin(0)=0\) (since \(\sin(0)\) is a basic trigonometric value)
• For \(x=\frac{\pi}{2}\), \(\sin(\frac{\pi}{2}) = 1\) (basic trigonometric value)

Step2: Construct the table for \(y = \sin^{-1}(x)\)

Since \(y=\sin^{-1}(x)\) is the inverse of \(y = \sin(x)\) on the interval \([-\frac{\pi}{2},\frac{\pi}{2}]\), if \(\sin(a)=b\), then \(\sin^{-1}(b)=a\)

• When \(x=-1\), \(\sin^{-1}(-1)=-\frac{\pi}{2}\)
• When \(x = 0\), \(\sin^{-1}(0)=0\)
• When \(x = 1\), \(\sin^{-1}(1)=\frac{\pi}{2}\)

Answer:

Part (a)

\(x\)	\(\sin(x)\)
\(0\)	\(0\)
\(\frac{\pi}{2}\)	\(1\)

Part (b)

\(x\)	\(\sin^{-1}(x)\)
\(0\)	\(0\)
\(1\)	\(\frac{\pi}{2}\)

For part (c), to plot the points:

• For \(y = \sin(x)\): the points are \((-\frac{\pi}{2},-1)\), \((0,0)\), \((\frac{\pi}{2},1)\)
• For \(y=\sin^{-1}(x)\): the points are \((-1,-\frac{\pi}{2})\), \((0,0)\), \((1,\frac{\pi}{2})\). Plot these points on the coordinate - axis where for \(y = \sin(x)\), \(x\) is on the horizontal axis (domain \([-\frac{\pi}{2},\frac{\pi}{2}]\)) and \(y\) is on the vertical axis (range \([-1,1]\)); for \(y=\sin^{-1}(x)\), \(x\) is on the horizontal axis (domain \([-1,1]\)) and \(y\) is on the vertical axis (range \([-\frac{\pi}{2},\frac{\pi}{2}]\))