Question

Sketch a graph of the given pair of functions to conjecture a relationship between the two functions. Then verify the conjecture. $$\sin ^{-1} x ; \frac{\pi}{2}-\cos ^{-1} x$$

Short answer

Answer: Graphically, the functions \(y=\sin^{-1}x\) and \(y=\frac{\pi}{2}-\cos^{-1}x\) are symmetrical over the line \(y = \frac{\pi}{4}\), meaning their graphs have a reflectional relationship. Algebraically, they are related through the equation \(\sin^{-1}x = \frac{\pi}{2} - \cos^{-1}x\).

Step-by-step solution

Graph the first function, \(\sin^{-1}x\)

To sketch the graph of the function \(y=\sin^{-1}x\), remember the domain of \(\sin^{-1}x\) is \([-1,1]\) and the range is \([-\frac{\pi}{2}, \frac{\pi}{2}]\). The sine inverse function has points of symmetry at \((0,0)\) and \((1,\frac{\pi}{2})\). Therefore, the graph should pass through these points, and will look like a smooth curve between them.

Graph the second function, \(\frac{\pi}{2}-\cos^{-1}x\)

To sketch the graph of the function \(y=\frac{\pi}{2}-\cos^{-1}x\), first recognize that it can be represented as the transformation of the cosine inverse function, \(\cos^{-1}x\). The domain of \(\cos^{-1}x\) is \([-1,1]\) and the range is \([0, \pi]\). The transformation \(\frac{\pi}{2}-\cos^{-1}x\) slides the graph of \(\cos^{-1}x\) down by \(\frac{\pi}{2}\) units and reflects it across the horizontal line \(y = \frac{\pi}{4}\). This means that the points of symmetry will be located at \((0,\frac{\pi}{2})\) and \((1,0)\). The graph should pass through these points and will look like a smooth curve between them.

Conjecture a relationship between the two functions

Upon sketching both graphs, you can observe that the two functions are symmetrical over the line \(y = \frac{\pi}{4}\). This means their graphs have a reflectional relationship. When we notice this symmetry, we can conjecture that the two functions are related through the equation: $$\sin^{-1}x = \frac{\pi}{2} - \cos^{-1}x$$

Verify the relationship algebraically

To verify the relationship, use the inverse trigonometric identity (specifically, the identity between sine inverse and cosine inverse): $$\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}$$ To confirm the relationship \(\sin^{-1}x = \frac{\pi}{2} - \cos^{-1}x\), substitute the right side of the conjectured equation into the trigonometric identity: $$\frac{\pi}{2} - \cos^{-1}x + \cos^{-1}x = \frac{\pi}{2}$$ Since both sides are equal, we can conclude that the conjecture is true and the relationship between the two functions is as follows: $$\sin^{-1}x = \frac{\pi}{2} - \cos^{-1}x$$

Key concepts

Domain and Range

Understanding the domain and range of inverse trigonometric functions is crucial. The function \(\sin^{-1}x\), known as arcsine, has a domain of \([-1, 1]\) because the sine of any angle must fall between these values. Its range is \([-\frac{\pi}{2}, \frac{\pi}{2}]\), meaning the output angles are restricted to these values. This ensures that \(\sin^{-1}x\) is a function, as different x-values don't map to the same angle.
For \(\frac{\pi}{2} - \cos^{-1}x\), we must first consider \(\cos^{-1}x\), whose domain is also \([-1, 1]\). The range of \(\cos^{-1}x\) is \([0, \pi]\), but with the transformation \(\frac{\pi}{2} - \theta\), the range becomes \([-\frac{\pi}{2}, \frac{\pi}{2}]\). This adjusted range perfectly aligns with \(\sin^{-1}x\), hinting at their mathematical relationship.

Graphical Symmetry

Graphical symmetry helps visualize relationships between functions and their inverses. When graphing \(\sin^{-1}x\) and \(\frac{\pi}{2} - \cos^{-1}x\), you'll notice symmetry across the line \(y = \frac{\pi}{4}\). This happens because when you reflect one of these graphs over this line, you get the other.

• The graph of \(\sin^{-1}x\) passes through points like \((0,0)\) and \((1,\frac{\pi}{2})\).


• The graph of \(\frac{\pi}{2} - \cos^{-1}x\) passes through \((0,\frac{\pi}{2})\) and \((1,0)\).

This symmetrical reflection confirms our earlier conjecture of their relationship: \(\sin^{-1}x = \frac{\pi}{2} - \cos^{-1}x\). Graphical symmetry isn’t just interesting; it's a visual confirmation of algebraic identities.

Trigonometric Identities

Trigonometric identities are powerful tools that can simplify complex expressions and help establish relationships between functions. With inverse trigonometric identities, the key one here is:
\[ \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \]
This identity shows how arcsine and arccosine are interconnected. By rearranging it, we easily see \(\sin^{-1}x = \frac{\pi}{2} - \cos^{-1}x\).

• This means that for every angle \(x\), the sum of its arcsine and arccosine equals \(\frac{\pi}{2}\).


• This confirms the complementary nature of sine and cosine.

These identities simplify problem-solving and give deeper insight into the harmonious relationships of trigonometric functions.