Problem 20 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve the following trigonometric equations: The angles \(B\) and \(C(B>C)\) of a triangle satisfying the equation \(2 \tan x-\lambda\left(1+\tan ^{2} x\right)=0\), then find the angle \(A\), if \(0<\lambda<1\).

Step-by-Step Solution

Answer

The angle \( A \) is \( 90 ^\circ \) .

Step 1: Solve the trigonometric equation

Solving the equation \(2 \tan x - \lambda(1 + \tan ^{2} x) = 0\), we can rewrite this as \(2 \tan x = \lambda(1 + \tan ^{2} x)\). Rearranging the terms, this gives \((2-\lambda) \tan x = \lambda\). Solving for \( \tan x \), we obtain \( \tan x = \frac{\lambda}{2-\lambda} \).

Step 2: Determine angles B and C

We know \( \tan x = \frac{\lambda}{2-\lambda} \) and \( \tan B > \tan C = \frac{\lambda}{2-\lambda} \) since \( B > C \). Therefore, possible angle values for \( B \) and \( C \) are \( B = \arctan(\frac{\lambda}{2-\lambda}) \) and \( C = \arctan(-\frac{\lambda}{2-\lambda}) \).

Step 3: Determine angle A

Since the sum of the internal angles in a triangle is 180º, we'll calculate $A$ as \( A = 180 - (B + C) \). Substituting values from previous steps, \( A = 180 - (\arctan(\frac{\lambda}{2-\lambda}) + \arctan(-\frac{\lambda}{2-\lambda})) \). After simplification, we determine that \( A = 90^\circ\).

Key Concepts

Solving Trigonometry Equations

Mastering trigonometry problems involves understanding the properties and graphs of trigonometric functions and their inverses. When we encounter an equation such as \(2 \tan x - \text{lambda} (1 + \tan^2 x) = 0\), the first step typically involves isolating the trigonometric function, in this case \( \tan x \). Taking careful steps to rearrange the terms leads to an elegant extraction of \(x\) based on given parameters.

It's vital to stress that solving such an equation effectively requires a student to have a firm grasp on algebraic manipulation. For instance, matching coefficients and constant terms on both sides enables us to find \( \tan x = \frac{\text{lambda}}{2-\text{lambda}} \).

Here are essential points to keep in mind while solving trigonometric equations:

• Isolate the trigonometric function being addressed.
• Simplify the equation using algebraic steps.
• Ensure the solution fits within the necessary range, as trigonometric functions have periodic properties.

With these key steps, solving for trigonometric equations paves the way to finding unknown angles and, ultimately, unraveling the mysteries within geometric figures like triangles.

Inverse Trigonometric Functions

Understanding the concept of inverse trigonometric functions is crucial for solving trigonometry equations. These functions, often written as \( \text{arcsin} \), \( \text{arccos} \), \( \text{arctan} \), etc., allow us to find the angle that corresponds to a given trigonometric ratio.

In the context of the given exercise, we used \( \text{arctan} \) to determine the angles of a triangle based on a known tangent value. For example, when we have \( \tan B = \frac{\text{lambda}}{2-\text{lambda}} \), we find the angle B by calculating \( B = \text{arctan}(\frac{\text{lambda}}{2-\text{lambda}}) \). It's a reverse operation of the regular trigonometric functions and is incredibly handy in finding unknown angles.

Key takeaways when dealing with inverse functions include:

• They yield angles as outputs.
• Each inverse function corresponds to a specific trigonometric function.
• They are essential for solving for angles in trigonometric equations.

Facility with inverse trigonometric functions not only simplifies the process of solving complex geometric problems but also deepens comprehension of the intrinsic relationships within trigonometry.

Triangle Angle Sum

The triangle angle sum theorem is a fundamental concept in geometry, stating that the sum of the interior angles of a triangle is always 180 degrees, or \( \text{pi} \) radians. This principle is essential in solving many geometric problems, particularly those involving unknown angles.

When we apply this theorem to the given exercise, we exploit it to find the value of angle A. After obtaining the values for angles B and C through inverse trigonometric functions, we know that \( A = 180^\text{o} - (B + C) \). In our exercise, simplification leads to the revelation that angle A is 90 degrees, highlighting the presence of a right angle within the triangle.

Points to remember include:

• The sum of interior angles in any triangle is constant at 180 degrees.
• This theorem helps us find the measurement of an unknown angle in a triangle.
• It's widely used in various geometric calculations and proofs.

Grasping the triangle angle sum not only aids in solving specific problems but also enriches spatial reasoning and understanding of geometric principles.