Problem 72 - A Complete Resource Book in Mathematics for JEE Main

Question

In a \(\Delta P Q R\), if \(3 \sin P+4 \cos Q=6\) and \(4 \sin Q+3 \cos\) \(P=1\), then the angle \(R\) is equal to (A) \(\frac{5 \pi}{6}\) (B) \(\frac{\pi}{6}\) (C) \(\frac{\pi}{4}\) (D) \(\frac{3 \pi}{4}\)

Step-by-Step Solution

Answer

The angle \(R\) equals \(\frac{5\pi}{6}\) (Option A).

Step 1: Identify Given Equations

The problem provides two equations based on trigonometric functions in triangle \(\Delta PQR\):\(3 \sin P + 4 \cos Q = 6\) and \(4 \sin Q + 3 \cos P = 1\). Our goal is to find angle \(R\).

Step 2: Express in Terms of Known Functions

Using the identities of sin and cos, express both equations in terms of square identities: \((\sin^2 P + \cos^2 P = 1)\) and \((\sin^2 Q + \cos^2 Q = 1)\). Find \(\sin P\) and \(\cos Q\) from the first equation, and \(\cos P\) and \(\sin Q\) from the second equation.

Step 3: Solve System of Equations

Solve the given simultaneous equations: \(3 \sin P + 4 \cos Q = 6\) and \(4 \sin Q + 3 \cos P = 1\). Substitute values from one equation into the other to find expressions for each trigonometric function.

Step 4: Calculate Angles

Determine \(P\) and \(Q\) using the solved trigonometric values from the system of equations. Compute \(\sin P\), \(\cos Q\), \(\sin Q\), \(\cos P\) using known trigonometric identities.

Step 5: Use Triangle Angle Sum

In any triangle, the sum of all interior angles is \(\pi\). Therefore, calculate \(R = \pi - P - Q\) using the angles derived from the trigonometric identities and solved equations.

Step 6: Select Correct Option

After calculating \(R\), compare it with the given options. Substitute back to verify if \(R\) satisfies the initial conditions.

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with the same set of unknowns. In this problem, we are given two trigonometric equations involving the angles of a triangle, \( \Delta PQR \):

• \( 3 \sin P + 4 \cos Q = 6 \)
• \( 4 \sin Q + 3 \cos P = 1 \)

These equations must be solved simultaneously to find values for \( \sin P, \cos Q, \sin Q, \) and \( \cos P \). The solution involves manipulating these expressions to isolate each trigonometric function. This allows us to solve for the unknown angles in the triangle. It is crucial to understand how to solve these systems of equations to find precise solutions for the trigonometric values involved.

Angle Sum Property

The angle sum property is a fundamental concept in triangle geometry. It states that the sum of the interior angles in a triangle always equals \( \pi \) radians, or 180 degrees. This property is particularly useful for triangles with trigonometric elements. After calculating angles \( P \) and \( Q \), we use the angle sum property to find angle \( R \):

• \( P + Q + R = \pi \)
• Solving for \( R \) yields \( R = \pi - P - Q \)

Using the calculated values from the earlier steps, apply this property to determine the measure of angle \( R \). This provides a complete understanding of the geometric properties of the triangle \( \Delta PQR \).

Trigonometric Identities

Trigonometric identities are equations that relate the trigonometric functions to one another. They simplify the process of solving objectives like finding angles or sides in a triangle. Important identities used in this exercise include:

• The Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)

These identities are useful for expressing one trigonometric function in terms of another, thus reducing the complexity of the system of equations. In this exercise, we utilize these identities to express equations purely in terms of known functions. This aids in effectively solving for \( \sin P \), \( \cos Q \), \( \cos P \), and \( \sin Q \) needed to determine the angles of triangle \( \Delta PQR \).

Triangle Geometry

Triangle geometry involves understanding the properties and relationships within a triangle. These properties include the sides, angles, and the types of triangles. For example, using trigonometry provides insights into the relationships between a triangle's angles and its side lengths.

• Each triangle has a unique set of angle and side length combinations that dictate its properties.
• Trigonometric functions help in finding unknown angles or sides when certain values are known.

In this problem, understanding the geometric properties of \( \Delta PQR \) is essential to solve for angle \( R \). Utilizing knowledge of triangle geometry ensures a deeper comprehension of why the system of equations and corresponding trigonometric identities efficiently reveal the angle measurements.