Problem 61 - A Complete Resource Book in Mathematics for JEE Main

Question

The sides of a triangle are \(\sin \alpha, \cos \alpha\) and \(\sqrt{1+\sin \alpha \cos \alpha}\) for some \(0<\alpha<\frac{\pi}{2}\). Then the greatest angle of the triangle is (A) \(60^{\circ}\) (B) \(90^{\circ}\) (C) \(120^{\circ}\) (D) \(150^{\circ}\)

Step-by-Step Solution

Answer

The greatest angle of the triangle is \(120^{\circ}\) (Option C).

Step 1: Identify the Sides of the Triangle

The sides of the triangle are given as \( \sin \alpha, \cos \alpha \), and \( \sqrt{1+\sin \alpha \cos \alpha} \). Let's assign these values to sides \( a, b, \) and \( c \), where \( c \) is the longest side, so \( c = \sqrt{1+\sin \alpha \cos \alpha} \).

Step 2: Determine the Type of Triangle

Since \( c \) is the longest side, we use the cosine rule to find the angle opposite side \( c \). The cosine rule states that for a triangle with sides \( a, b, \) and \( c \), and angle \( \gamma \) opposite the side \( c \): \[c^2 = a^2 + b^2 - 2ab \cos \gamma\].

Step 3: Calculate \(c^2\) and \(a^2 + b^2\)

Calculate \( c^2 = (1 + \sin \alpha \cos \alpha) \). Also, calculate \( a^2 + b^2 = \sin^2 \alpha + \cos^2 \alpha = 1\).

Step 4: Find \( \cos \gamma \) Using the Cosine Rule

Substitute the values into the cosine rule: \[ 1 + \sin \alpha \cos \alpha = 1 - 2\sin\alpha \cos\alpha \cos \gamma \] Solve for \( \cos \gamma \): \[ \sin \alpha \cos \alpha + 2\sin\alpha \cos\alpha \cos \gamma = 0 \] \[ \cos \gamma = -\frac{1}{2} \].

Step 5: Determine the Greatest Angle

The cosine of angle \( \gamma \) is \( -\frac{1}{2} \), which corresponds to an angle \( \gamma = 120^{\circ} \). Since \( \gamma \) is the angle opposite the longest side, it is indeed the greatest angle of the triangle.

Key Concepts

Cosine Rule

The Cosine Rule is a fundamental tool in trigonometry for finding an unknown angle or side in a triangle. It's especially useful in non-right triangles, where right-angle trigonometric functions may not directly apply.
The cosine rule is given by:\[c^2 = a^2 + b^2 - 2ab\cos\gamma\]where:

• \(a\), \(b\), and \(c\) are the lengths of the sides of the triangle.
• \(\gamma\) is the angle opposite to side \(c\).

This rule is a direct extension of the Pythagorean theorem, which applies to right triangles.
The cosine rule becomes necessary when angles or sides need to be determined for triangles that are not necessarily right-angled.
In this exercise, we used the cosine rule to find the largest angle in a triangle given the lengths of all three sides.

Triangle Properties

Understanding the properties of triangles is crucial in solving geometry problems. A triangle always has three sides and three angles, and some key properties apply to all triangles.
Firstly, the sum of the interior angles of a triangle is always \(180^\circ\). This helps in checking the accuracy of calculated angles. Furthermore, the largest angle in a triangle is always opposite the longest side.
The triangle in this exercise has sides defined in terms of trigonometric functions, which showcases another important aspect—the relationships between angles and the corresponding sides.

• When analyzing a triangle, ensure that the sides and angles fit within these fundamental properties to confirm the solution's validity.
• The use of trigonometric identities, such as \(\sin^2 \alpha + \cos^2 \alpha = 1\), also plays a crucial role in simplifying calculations.

Angle Calculation

Calculating angles in a triangle can be tackled with different methods, often using trigonometric identities and rules. In scenarios involving non-right triangles, the cosine rule facilitates this process.
For the given exercise, the largest angle, \(\gamma\), opposite the longest side was determined.
By substituting known values into the cosine rule and simplifying the equation, we arrived at:\[\cos \gamma = -\frac{1}{2}\]This corresponds to an angle \(\gamma = 120^\circ\).

• The negative sign indicates that the angle is obtuse, which is expected for the largest angle opposite the longest side.
• Understanding how to manipulate these calculations through algebraic simplification and recognizing known trigonometric values is essential to proficiently solving such problems.