Problem 9 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Let \(m=\sin ^{-1}\left(a^{6}+1\right)+\cos ^{-1}\left(a^{4}+1\right)-\tan ^{-1}\left(a^{2}+1\right)\), then find the image of the line \(x+y=m\) about the \(y\)-axis.

Step-by-Step Solution

Answer

The image of the line \(x + y = m\) about the y-axis is \(y = x + m\). This is achieved by changing the sign of all 'x' coordinates in the original equation to get the equation for the mirrored line.

Step 1: Trigonometric Expression

Firstly, one can observe that there is a given expression \( m = sin^{-1}(a^{6}+1) + cos^{-1}(a^{4}+1) - tan^{-1}(a^{2}+1)\). This expression merely provides a common value for all the inverse trigonometric expressions. The inverse trigonometric functions are real-number functions, so they have to be cast into a form applicable for the problem. Thus, a new expression for 'm' gives us a line in terms of x and y.

Step 2: Equation of the Line

The given equation of the line is \(x + y = m\). In order to plot this line and consequently mirror it about the y-axis, it's best to write it in slope-intercept form (y = mx + b). Subtracting 'x' from both sides of the equation gives \(y = -x + m\)

Step 3: Reflect the Line about the y-axis

The line \(y = -x + m\) has all its 'x' coordinates changed to their opposites when mirrored about the y-axis. That means, the image of the line about the y-axis has the equation \(y = x + m\)

Key Concepts

Reflection of Lines

Understanding the reflection of lines can be a helpful visualization exercise in coordinate geometry. When we talk about reflecting a line, we usually mean flipping it over one of the coordinate axes. Here, we reflect the line about the y-axis.

Reflecting a line across the y-axis involves changing the x-coordinates of every point on the line to their opposites. To illustrate, if we have a line described by the equation \(y = -x + m\), its reflection across the y-axis results in a new line equation \(y = x + m\).

This means:

• Positive x-values become negative.
• Negative x-values become positive.

By simply changing every x in the equation to \(-x\), we reflect the entire line, keeping the y-intercept the same. Be sure to practice this by sketching lines and their reflections on graph paper to bolster your understanding.

Coordinate Geometry

Coordinate geometry, often called analytic geometry, allows us to represent geometric shapes using coordinates and algebraic equations. It's like a bridge connecting algebra and geometry, where we can express geometric problems in a visual way.

Key components of coordinate geometry include:

• Points: Given as coordinates \((x, y)\).
• Lines: Typically represented as equations like \(y = mx + b\), where \(m\) is the slope.
• Equation of a Line: The equation \(x + y = m\) transforms to \(y = -x + m\) for simplicity.

Reflection, as seen in coordinate geometry, emphasizes spatial relationships and symmetry. For students, understanding these relationships helps in visualizing solutions to algebraic expressions easily. Applying these concepts can make tackling complex word problems a breeze and bring clarity to abstract ideas.

Trigonometric Identities

Trigonometric identities involve relationships among trigonometric functions that hold true for all values within their domains. In the context of inverse trigonometric functions, these identities help simplify complex expressions.

Some important inverse trigonometric functions are:

• \(\sin^{-1}(x)\)
• \(\cos^{-1}(x)\)
• \(\tan^{-1}(x)\)

These functions "undo" what the standard trigonometric functions do, leading us back from a ratio to an angle.

The exercise involves these functions:

• \(m = \sin^{-1}(a^6 + 1) + \cos^{-1}(a^4 + 1) - \tan^{-1}(a^2 + 1)\)

This expression helps us calculate a constant \(m\), which we then utilize in our line equation \(x + y = m\). Trigonometric identities simplify such expressions, making the evaluation more straightforward. Understanding these concepts is crucial for solving higher-level math problems efficiently.