Question

In any cyclic quadrilateral \(\mathrm{ABCD}, \cos \mathrm{A}+\cos \mathrm{B}+\cos \mathrm{C}+\cos \mathrm{D}\) is equal to (a) 0 (b) 1 (c) \(-1\) (d) 2

Short answer

Answer: (a) 0

Step-by-step solution

Recall the properties of cyclic quadrilaterals

A quadrilateral is cyclic if its vertices lie on a circle. One property of cyclic quadrilaterals is that the sum of a pair of opposite angles is 180 degrees, i.e., \(\angle A + \angle C = 180^\circ\), and \(\angle B + \angle D = 180^\circ\).

Find the sum of cosines

We want to find the value of \(\cos A + \cos B + \cos C + \cos D\). We know that \(A + C = 180^\circ\) and \(B + D = 180^\circ\). Since \(\cos (180^\circ - x) = -\cos x\), we can rewrite the sum as follows: \(\cos A + \cos B + \cos C + \cos D = \cos A + \cos B + \cos (180^\circ - A) + \cos (180^\circ - B)\) Using the property \(\cos (180^\circ - x) = -\cos x\), we get: \(\cos A + \cos B - \cos A - \cos B = 0\)

Conclusion

The sum of the cosines of the angles of a cyclic quadrilateral ABCD is equal to 0. Therefore, the correct answer is (a) 0.

Key concepts

Mathematics IIT-JEE

The Indian Institutes of Technology Joint Entrance Examination (IIT-JEE) is a highly competitive exam known for its challenging mathematics section. One topic that often appears in this section deals with the properties and applications of cyclic quadrilaterals. Understanding cyclic quadrilaterals is vital as they frequently form the basis for complex geometric proofs and problems. Such questions test a student's ability to integrate knowledge of various geometrical concepts and to apply them in a cohesive manner. Given the depth of this topic, IIT-JEE aspirants need to master the related concepts thoroughly, often requiring a solid grasp of not just the properties of cyclic quadrilaterals but also trigonometry, as in the case of finding the cosine of angles within these quadrilaterals.

To excel in mathematics in the IIT-JEE, a student should be adept at interpreting geometric figures, understanding theorems, and applying these to arrive at solutions quickly and accurately. Exercises such as the one provided are practical applications of such skills and are crucial for success in the exam.

Cosine of Angles in Quadrilaterals

Understanding the cosine of angles in quadrilaterals, especially cyclic quadrilaterals, is a core component of trigonometry and geometry that can lead to solving many complex problems. The cosine function relates the angle of a triangle to the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle. However, its application extends beyond right triangles.

In a cyclic quadrilateral, since the opposite angles are supplementary, that is, they add up to 180 degrees, the concept of cosine can be used ingeniously. If one angle is \(x\), the other is \(180^\textdegree - x\), and using the trigonometric identity \(\cos(180^\textdegree - x) = -\cos(x)\), we can deduce that the cosines of supplementary angles are negatives of each other. This property simplifies calculations and is beneficial in various trigonometric proofs and solutions in mathematics.

Properties of Cyclic Quadrilaterals

Cyclic quadrilaterals have some fascinating properties that make them unique. A quadrilateral is cyclic if a single circle passes through all four of its vertices. One of its most essential properties is that opposite angles add to 180 degrees, making them supplementary. As a consequence, this means that the sine of opposite angles is equal, and the cosine of one is the negative of the other.

Furthermore, in cyclic quadrilaterals, there are significant relations involving their sides and angles, such as Ptolemy's theorem, which can be used to find the product of the lengths of the diagonals. Knowing these properties allows for a breadth of geometric exercises to be tackled with precision, which is paramount for students, especially when engaging in competitive exams like the IIT-JEE that often feature problems requiring an in-depth understanding of cyclic quadrilaterals.