If \(2+12 \cos \theta-16 \cos ^{3} \theta=A\), then \(A\) lies in the interval is (a) \([-2,-1]\) (b) \([-2,1]\) (c) \([-6,2]\) (d) \([-2,6]\)

The cosine function is a fundamental concept in trigonometry, a field of mathematics that studies the relationships between the angles and sides of triangles. It is denoted as cos(θ) where θ is the angle in question, and it maps the angle to a real number between -1 and 1. This function is periodic, with a period of 2π radians (or 360 degrees), meaning that the function values repeat every full rotation around a circle.

The cosine function has its maximum value of 1 when θ is an integer multiple of 2π, and its minimum value of -1 when θ is an odd multiple of π. Knowing these extrema is crucial when trying to determine the maximum and minimum values of any expression involving the cosine function, such as in the JEE Maths problem we're discussing.

Properties of the Cosine Function

• It is an even function: cos(-θ) = cos(θ).
• The cosine of 0 degrees (or 0 radians) is 1, representing the maximum value of the function.
• It has its zeros at θ = π/2 + kπ, where k is an integer.

These characteristics are essential in solving many trigonometric equations and problems, including those commonly found in JEE Maths exams.

Interval notation is a mathematical way of representing a range of numbers along a number line. It provides a concise method for showing the beginning and ending of a range. In interval notation, brackets are used to indicate closed intervals where the endpoints are included, while parentheses indicate open intervals where the endpoints are not included.

For example, the interval [-1, 1] represents all numbers from -1 to 1, including -1 and 1. On the other hand, (-1, 1) would represent all numbers between -1 and 1 exclusively, without including the endpoints. The concept becomes especially useful in calculus and algebra, where it is often necessary to specify the domain and range of functions.

Key Points in Interval Notation

• [a, b]: Closed interval, includes both a and b.
• (a, b): Open interval, excludes both a and b.
• [a, b) or (a, b]: Half-open interval, includes one endpoint but not the other.
• Infinity symbols are always paired with parentheses, as they are not real numbers and cannot be included in the interval (e.g., (-∞, a] or [b, ∞)).

Understanding this notation is vital for correctly interpreting the ranges in calculus, algebra, and, as seen here, in solving JEE Maths problems.

Trigonometric identities are equations that involve trigonometric functions and are true for all values within their domains. These identities are incredibly useful for simplifying expressions and solving trigonometric equations. A thorough knowledge of these can greatly benefit students tackling JEE Maths problems, as it allows for the transformation of complex equations into more manageable forms.

Some of the most fundamental trigonometric identities include the Pythagorean identities, like cos^2(θ) + sin^2(θ) = 1, and angle sum and difference identities, such as cos(α ± β) = cos(α)cos(β) ∓ sin(α)sin(β). These can be used to find the values of trigonometric functions for various angles and to simplify trigonometric expressions.

Common Trigonometric Identities

• Reciprocal identities: sin(θ) = 1/csc(θ), cos(θ) = 1/sec(θ), and others.
• Quotient identities: tan(θ) = sin(θ)/cos(θ), cot(θ) = cos(θ)/sin(θ).
• Pythagorean identities: tan^2(θ) + 1 = sec^2(θ), 1 + cot^2(θ) = csc^2(θ).

Utilizing these identities correctly can lead to accurate and simplified solutions in trigonometry, which is an integral part of the mathematics curriculum for competitive exams like the JEE.