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

Question

In a \(\Delta A B C, \cos A+\cos B+\cos C=\frac{3}{2}\), then the \(\Delta\) (a) Isosceles (b) right angled (c) equilateral (d) None

Step-by-Step Solution

Answer

The given triangle is an isosceles triangle (Option (a)).

Step 1: Checking the nature of the triangle

The sum of the angles of any triangle is 180 degrees. For a particular \( \Delta ABC \), it is given that \( \cos A + \cos B + \cos C = \frac{3}{2} \). We know that the cosine of an angle is maximum when the angle is 0 and its value is 1. Since there are 3 angles, the max value the sum of cosines can take is 3 when all 3 angles are 0. But this is not possible in a triangle as the sum of angles in a triangle is 180° not 0°.

Step 2: Eliminating options and concluding

It means that the sum of cosines of A, B, C cannot be more than 1 each. But it is given that \( \cos A + \cos B + \cos C = \frac{3}{2} \), this is only possible if one of the angles is 60° and the other two angles are 0°. So the answer should be an isosceles triangle where two angles are equal i.e., base angles are equal. Thus, we can eliminate the right-angled triangle, as the cosine of 90 degrees is 0, and the equilateral triangle, as the cosine of 60 degrees is 1/2, and the sum of three cosines would only be 3/2. Hence, it's impossible to be these triangle types. With these eliminations we can reply the shape of the triangle is an isosceles triangle (option a) because two cosines need to be 1 (corresponds to 0 degrees) and one cosine needs to be -1/2 (corresponds to 120 degrees).

Key Concepts

Triangle properties

Understanding triangle properties is fundamental in solving problems related to them. A triangle, a three-sided polygon, has several distinctive properties:

• Sum of Angles: The sum of the internal angles in any triangle is always 180 degrees. This helps in understanding the relation between the angles.
• Side-Lengths: Any side can be described by its opposite angle. Longer sides generally have larger opposite angles.
• Angle Types: Angles in a triangle can vary; they're often acute (less than 90°), obtuse (more than 90°), or right (exactly 90°), affecting the triangle's classification.

Recognizing these properties allows us to draw specific conclusions, such as identifying if a triangle is equilateral, isosceles, or another type based on the behavior of its angles and sides.

Cosine rule

The cosine rule is a powerful tool in trigonometry, aiding in the solution of triangles, especially when other formulas like the sine rule aren't applicable. It can be expressed as:\[ c^2 = a^2 + b^2 - 2ab \cos C \]This rule applies to all triangles, not just right-angled ones, allowing us to:

• Find an unknown side when we know two sides and the included angle.
• Discover an angle when we know all three sides.

The cosine of an angle, particularly in an angular sum scenario like \( \cos A + \cos B + \cos C \), allows us to reveal the nature of a triangle without solving all angles or sides entirely. Recognizing that the cosine value can only reach 1 when the angle is 0 is crucial here, highlighting why our problem suggests a special configuration of angles.

Isosceles triangle

An isosceles triangle has two sides of equal length, leading to two angles being equal. This property emerges naturally from:

• Triangle Symmetry: Helping us conclude certain properties without calculating every angle or side.
• Angle Equality: The equal sides ensure the base angles are equal, simplifying some trigonometric relations.

In the given problem, when the sum of cosines doesn't align with typical triangle results, we can ascertain an isosceles configuration. This is due to two angles being equally acute or obtuse, leaving the third angle significantly different; yet, overall the properties of sum facing constraints by 180 degrees hold true. Understanding these properties is crucial in verifying and concluding triangle classifications based on given angle or cosine sum configurations.