Problem 75 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$\arctan x+\arctan \frac{1}{x}=\frac{\pi}{2}, x>0$$

Step-by-Step Solution

Answer

The identity holds true for all \(x > 0\).

Step 1: Understand the equation

The expression involves an identity that needs to be verified: \(\arctan x + \arctan \frac{1}{x} = \frac{\pi}{2}\). We need to show that for all \( x > 0 \), this equality holds true.

Step 2: Recall ArcTan Properties

Recall the property of arctangent: \( \arctan a + \arctan b = \frac{\pi}{2} \) if \( a \cdot b = 1 \) and \( a, b > 0 \). In this case, if we set \( a = x \) and \( b = \frac{1}{x} \), this product indeed equals 1.

Step 3: Use arctangent identity

Since \( x \cdot \frac{1}{x} = 1 \), we apply the identity: \( \arctan a + \arctan b = \frac{\pi}{2} \) when \( a \cdot b = 1 \). This confirms that \( \arctan x + \arctan \frac{1}{x} = \frac{\pi}{2} \) is true.

Key Concepts

Arctangent Function

The arctangent function, often denoted as \( \arctan(x) \), is the inverse of the tangent function. This means it finds the angle whose tangent is the number \( x \). It is specifically useful in trigonometry for determining angles when given the ratio of the opposite side to the adjacent side in a right triangle.

Some important properties of the arctangent function include:

• Its range is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), meaning it outputs angles between these values.
• It is an odd function, meaning \( \arctan(-x) = -\arctan(x) \).

Relation to Trigonometric Identities

Understanding how the arctangent function relates to other identities can greatly enhance problem-solving skills. For instance, knowing that \( \arctan(a) + \arctan(b) = \frac{\pi}{2} \) when \( a \cdot b = 1 \) helps in simplifying complex expressions involving angles.

Always remember: the arctangent tells you the angle whose tangent is a specific value, making it a vital tool in solving many trigonometric problems.

Angle Addition Formulas

Angle addition formulas are a set of identities in trigonometry that allow us to find the trigonometric values of angles summed together. These formulas are essential for evaluating trigonometric functions of angles that are not easily found using basic function tables or calculators.

For trigonometric operations, the angle addition formulas are:

• \( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \)
• \( \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \)
• \( \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)} \)

Applying Angle Addition to Arctangent

With the specific identity from the problem, \( \arctan(x) + \arctan\left(\frac{1}{x}\right) = \frac{\pi}{2} \), we are actually using an application of the tangent angle addition formula:

In this case, if \( a \cdot b = 1 \), the tangent part \( \tan(a + b) \) becomes undefined, because it corresponds to an angle of \( \frac{\pi}{2} \). This links back to the key understanding from the problem, bridging angle addition and the arctangent function.

Precalculus Concepts

Precalculus serves as a foundation for understanding various mathematical concepts, including trigonometry, which is heavily used in both calculus and physics.

In precalculus, students explore a wide range of functions, identities, and their properties, preparing them for more challenging courses. Trigonometry is a significant part of precalculus, involving the study of triangles, the unit circle, and how different functions relate to each other.

Importance of Trigonometric Identities in Precalculus

Using identities like \( \arctan(x) + \arctan\left(\frac{1}{x}\right) = \frac{\pi}{2} \) helps students understand deeper relationships between functions. It also enables solving practical problems like determining the heights of objects using angles and distances in real life.

Learning to verify identities is a crucial skill in precalculus, ensuring a solid grasp of how mathematical principles work, which is fundamental for progressing to calculus. Building a strong base in these concepts can simplify the transition to more complex topics later on.