Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \arcsin ^{2} x+\arccos ^{2} x=1 $$

Short answer

The given statement is false. The Pythagorean identity \(\cos^{2}\alpha + \sin^{2}\alpha=1\) applies to the values of the trigonometric functions of an angle, not to the angles themselves that result from inverse trigonometric functions. An example that proofs this is: \(\arcsin^2(0.5) + \arccos^2(0.5) = \pi^2/8\), which is not equal to 1.

Step-by-step solution

Understanding the arc functions

Arccosine is the inverse function of cosine. It returns the angle whose cosine is the specified number. On the other hand, arcsine is the inverse function of sine, returns the angle whose sine is the specified number. It's important to remember that for any angle \( \alpha \), \(\cos^{2}(\alpha) + \sin^{2}(\alpha)=1\), but this doesn't necessarily apply to their inverse functions.

Evaluate the given equation

We are asked if \(\arcsin^{2} x + \arccos^{2} x = 1 \). Replacing x by any valid value, let's say 0.5, we have: \(\arcsin^2(0.5) + \arccos^2(0.5) = \pi^2/16+\pi^2/16= \pi^2/8 \) which is not equal to one. Therefore the statement is false.

Explain why or give an example

The initial equation is false because the Pythagorean identity \(\cos^{2}\alpha + \sin^{2}\alpha=1\) refers to the values of the trigonometric functions of an angle, not to the angles themselves that result from an inverse trigonometric function like arcsin or arccos. As an example, we evaluated the statement with x being equal to 0.5, where the result wasn't equal to one, thus showing the statement is false.

Key concepts

Arcsine and Arccosine

The terms 'arcsine' and 'arccosine' refer to inverse trigonometric functions, which are essential for finding the angles if the value of the sine or cosine is known. When we write \(\arcsin(x)\), we're seeking the angle whose sine is \(x\), and it's important to note that the angle must be in the range \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). Similarly, \(\arccos(x)\) finds the angle whose cosine is \(x\) within the range of 0 to \(\pi\).

It's a common misconception to think that these functions can be squared and added to get 1, as shown in the false statement \(\arcsin ^{2} x+\arccos ^{2} x=1\). This mistake stems from confusing the functions with the Pythagorean identity. Let's move on to demystify that identity.

Pythagorean Identity

The Pythagorean identity is a fundamental equation in trigonometry, expressed as \(\sin^{2}(\theta) + \cos^{2}(\theta) = 1\), where \(\theta\) is an angle measured in radians. This identity is derived from the Pythagorean theorem in a right-angled triangle and is valid for all angles.

When considering inverse trigonometry, it's crucial to understand that the Pythagorean identity applies to the sine and cosine of an angle, not the angles obtained from the arcsine or arccosine functions. Thus, the identity doesn't hold when we're dealing with the squares of arcsine or arccosine values.

Trigonometric Functions

Trigonometric functions, including sine, cosine, and tangent, relate the angles of a triangle to the ratios of its sides. They are pivotal in various fields such as physics, engineering, and mathematics. These functions are periodic, and each has a specific range of possible values.

Being familiar with these functions assists in understanding their inverses - arcsine, arccosine, and arctangent. By definition, inverse trigonometric functions will always yield an angle, which can lead to confusion when approaching equations or proofs without clear distinction between angles and ratios.

Trigonometry Proofs

Trigonometry proofs require a clear understanding of trigonometric identities and relationships. These proofs often involve demonstrating that one side of an equation is equal to the other, or that a given statement is true or false, using logical reasoning and trigonometric concepts.

For instance, disproving the statement \(\arcsin ^{2} x+\arccos ^{2} x=1\) involves recognizing that it incorrectly applies the Pythagorean identity, which refers to the values of the trigonometric functions, not the angles given by the inverse functions. To correctly approach proofs, it's crucial to know the properties and domains of each trigonometric function and their inverses.