Problem 22 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

The value of \(\tan ^{-1}\left(\frac{a}{b+c}\right)+\tan ^{-1}\left(\frac{b}{a+c}\right)\), if \(\angle C=90\), in triangle \(A B C\) is (a) \(\frac{\pi}{4}\) (b) \(\frac{\pi}{3}\) (c) \(\frac{\pi}{2}\) (d) \(\pi\)

Step-by-Step Solution

Answer

\(\frac{\pi}{4}\)

Step 1 - Apply the right triangle property

Since \(C\) is \(90^{\circ}\), the triangle \(ABC\) is a right triangle. This allows us to apply the Pythagorean theorem which states that the square of the hypotenuse (side opposite the right angle) is equal to the sum of the squares of the other two sides. Here, if we take \(c\) as the hypotenuse, then \(c^2 = a^2 + b^2\).

Step 2 - Use inverse tangent addition formula

The addition formula for inverse tangent is defined as: \(\tan^{-1}\left(\frac{x+y}{1-xy}\right) = \tan^{-1}(x) +\tan^{-1}(y)\). So we can express the given expression \(\tan^{-1}\left(\frac{a}{b+c}\right)+\tan^{-1}\left(\frac{b}{a+c}\right)\) as \(\tan^{-1}\left(\frac{a+\frac{b}{a+c}}{1-\frac{a*b}{(b+c)*(a+c)}}\right)\).

Step 3 - Simplify the expression

After substituting \(c^2 = a^2 + b^2\) from Step 1 into the formula from Step 2, simplifying the expression becomes easier. Once simplified, the expression becomes \(\tan^{-1}(1)\), or in angle form, \(\frac{\pi}{4}\).

Key Concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in the field of geometry that provides a relationship between the sides of a right-angled triangle. The theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

This can be expressed algebraically as: \[ c^2 = a^2 + b^2 \] where \(c\) denotes the length of the hypotenuse, and \(a\) and \(b\) stand for the lengths of the triangle's other two sides.

For students working through geometry problems, understanding and applying the Pythagorean theorem is critical in order to calculate distances, prove right triangle properties, or solve problems involving two-dimensional figures.

Right Triangle Properties

Right triangles are characterized by one angle being exactly 90 degrees, and as a result of this, they exhibit several interesting properties that make them particularly important in geometry and trigonometry. Besides the Pythagorean theorem, other properties of right triangles include the relationships between their angles and sides:

• The two non-right angles in a right triangle always sum up to 90 degrees.
• The side lengths are proportional to the values of trigonometric functions for the angles. These functions are sine, cosine, and tangent.
• The hypotenuse is always the longest side of the triangle.

When working with inverse trigonometric functions, it’s useful to visualize them in terms of right triangles. For example, \(tan^{-1}(x)\) represents the angle whose tangent is \(x\). This visualization aids greatly in understanding the solutions to trigonometric problems.

Tangent Addition Formula

The tangent addition formula is powerful in solving trigonometric expressions involving the sum of two angles. The formula states that for any real numbers \(x\) and \(y\), provided \(1-xy\) is not zero, we have:

\[ \tan^{-1}\left(\frac{x+y}{1-xy}\right) = \tan^{-1}(x) + \tan^{-1}(y) \]

Applying this formula, one can often simplify complex trigonometric expressions by reducing a sum of inverse tangents to a single inverse tangent. This is particularly useful in equations where direct application of the inverse tangent to each term is not practical or possible. By recognizing when and how to apply the tangent addition formula, students can simplify their work and solve problems that involve adding angles more efficiently.