Question

Verifying identities 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

Question: Prove that the following relationship holds for all x in the domain $$[-1,1]$$: $$\sin^{-1} x = \frac{\pi}{2} - \cos^{-1} x$$ Answer: By taking the cosine of both sides of the equation and applying the trigonometric identity $$\cos(\alpha - \beta) = \cos(\alpha) \cos(\beta) + \sin(\alpha) \sin(\beta)$$, we get $$\cos(\sin^{-1} x) = \cos(\frac{\pi}{2}) \cos(\cos^{-1} x) + \sin(\frac{\pi}{2}) \sin(\cos^{-1} x)$$. Using the values of sine and cosine, we have $$\cos(\sin^{-1} x) = 0\cdot x + 1\cdot\sin(\cos^{-1} x)$$. From the Pythagorean identity, we find $$\sin(\cos^{-1} x) = \sqrt{1-x^2}$$. Plugging this value into the equation, we get $$\cos(\sin^{-1} x) = \sqrt{1-x^2}$$. Since the left-hand side is equal to the right-hand side for all x in the domain $$[-1,1]$$, the relationship is proven.

Step-by-step solution

Sketch the graphs of the functions

First, we will sketch the graph of the two given functions: 1. $$y = \sin^{-1} x$$ 2. $$y = \frac{\pi}{2} - \cos^{-1} x$$ For graphing, note that the domain of the inverse trigonometric functions is determined by the range of their corresponding trigonometric functions. In this case, the domain of both inverse sine and inverse cosine functions will be $$[-1,1]$$, and their range will be $$[0,\pi]$$. You can either sketch these graphs by hand or use a graphing tool such as Desmos, Geogebra, or any other online graphing calculator.

Conjecture a relationship between the two functions

When you sketch the graphs of both functions over the domain $$[-1,1]$$, you will notice that their graphs closely mirror each other. Based on this observation, we can conjecture that the two functions may have a relationship like this: $$\sin^{-1} x = \frac{\pi}{2} - \cos^{-1} x$$

Verify the conjectured relationship

Now we want to verify if this conjectured relationship is true for all x in the domain $$[-1,1]$$. We will use the properties of inverse trigonometric functions, and the Pythagorean identity for sine and cosine. To prove this identity, let's take the cosine of both sides: $$\cos(\sin^{-1} x) = \cos(\frac{\pi}{2} - \cos^{-1} x)$$ Next, we use the trigonometric identity $$\cos(\alpha - \beta) = \cos(\alpha) \cos(\beta) + \sin(\alpha) \sin(\beta)$$, where $$\alpha = \frac{\pi}{2}$$ and $$\beta = \cos^{-1} x$$: $$\cos(\sin^{-1} x) = \cos(\frac{\pi}{2}) \cos(\cos^{-1} x) + \sin(\frac{\pi}{2}) \sin(\cos^{-1} x)$$ Now, plug in the values of cosine and sine: $$\cos(\sin^{-1} x) = 0\cdot x + 1\cdot\sin(\cos^{-1} x)$$ Recall that sine and cosine are related by the Pythagorean identity $$\sin^2(\theta) + \cos^2(\theta) = 1$$. In this case, we have $$\sin^2(\cos^{-1} x) + \cos^2(\cos^{-1} x) = 1$$. Since $$\cos(\cos^{-1} x) = x$$, we get $$\sin^2(\cos^{-1} x) + x^2 = 1$$. This last equation implies that $$\sin(\cos^{-1} x) = \sqrt{1-x^2}$$. Now, we can plug in this value into our previous equation: $$\cos(\sin^{-1} x) = \sqrt{1-x^2}$$ Since the left-hand side is equal to the right-hand side for all x in the domain $$[-1,1]$$, the conjectured relationship is proven: $$\sin^{-1} x = \frac{\pi}{2} - \cos^{-1} x$$

Key concepts

Trigonometric Identities

Trigonometric identities are like puzzles in mathematics that help us simplify and solve trigonometric equations. They allow us to transform expressions into different forms, often making them easier to understand or compute.

Imagine having different tools that you can use to manipulate and simplify an equation—that’s what trigonometric identities offer! Here are some key identities you might encounter:

• Pythagorean Identities: These relate the square of sine, cosine, and tangent functions, such as \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
• Angle Sum and Difference Identities: These allow you to find the sine, cosine, or tangent of two angles added together or subtracted, like \( \cos(\alpha + \beta) = \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta) \).
• Reciprocal Identities: These connect a function to its reciprocal, such as \( \sin(x) = \frac{1}{\csc(x)} \).

By understanding these identities, you become proficient in manipulating and interpreting expressions and equations involving trigonometric functions. For example, they often enable you to prove equations like \( \sin^{-1} x = \frac{\pi}{2} - \cos^{-1} x \). The more you practice, the better you'll get at choosing and applying the right identity for any given problem.

Graphing Functions

Graphing functions is a visual way to understand the behavior of mathematical functions. It allows us to see patterns, trends, and relationships at a glance. When we graph functions, it becomes easier to perceive changes over different intervals and to make conjectures about the functions.

To graph a function, you plot points that represent the function and connect them to form a curve. For trigonometric functions, this involves recognizing their periodic nature and specific characteristics:

• Domain: The set of all possible input values ("x" values) for which the function is defined. In our case, both \( \sin^{-1} x \) and \( \frac{\pi}{2} - \cos^{-1} x \) have the domain \([-1, 1]\).
• Range: The set of all possible output values ("y" values) the function can take. Again, both functions in our example share the range \([0, \pi]\).
• Symmetry: Recognizing any symmetrical properties of the graph, like reflecting over an axis or around a point.

Graphing can be done by hand or with the help of graphing software, which is particularly useful for complex functions. Software such as Desmos or Geogebra can help illustrate dynamic changes instantly, and are an excellent aid in verifying conjectures, like when the graphs of \( \sin^{-1} x \) and \( \frac{\pi}{2} - \cos^{-1} x \) mirror each other.

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry, connecting the sine and cosine of an angle through their squares. It originates from the Pythagorean theorem in geometry and applies to trigonometric functions too.

The most familiar form of the Pythagorean identity is given by:
\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]

This formula indicates that for any angle \(\theta\), the sum of the square of the sine and the square of the cosine equals one. It serves as an invaluable tool for simplifying trigonometric expressions and proving relationships, especially when angles are involved.

For example, if you know \( \sin(\theta)\), you can use this identity to find \( \cos(\theta)\):

• If \( \sin(\theta) = a \), then \( \cos^2(\theta) = 1 - a^2 \), which implies \( \cos(\theta) = \sqrt{1 - a^2} \).

This relationship is crucial in verifying identities like \( \sin^{-1} x = \frac{\pi}{2} - \cos^{-1} x \), as it underpins the relationship between sine and cosine of angles that are inversely related. Understanding this identity not only helps with solving equations but also forms the foundation for more complex identities and problems.