Trigonometric Functions

In a unit circle, measure of angle subtended at the centre by an arc of length 1 unit is 1 radian. Domain of sin x and cos x is the set of all real numbers and range is the interval [–1, 1]. The domain of cosec x is the set { x : x ∈ R and x ≠ nπ, n ∈ Z} and range is the set {y : y ∈ R, y ≥ 1 or y ≤ – 1}. The domain of sec x is the set {x : x ∈ R and x ≠ (2n + 1)π/2, n ∈ Z} and range is the set {y : y ∈ R, y ≤ – 1 or y ≥ 1}. The domain of tan x is the set {x : x ∈ R and x ≠ (2n + 1)π/2, n ∈ Z} and range is the set of all real numbers. The domain of cot x is the set {x : x ∈ R and x ≠ nπ, n ∈ Z} and the range is the set of all real numbers. As x increases from 0 to π/2, sin x increases from 0 to 1. As x increases from π/2 to π, sin x decreases from 1 to 0. As x increases from π to 3π/2, sin x decreases from 0 to –1 and finally, sin x increases from –1 to 0 as x increases from 3π/2 to 2π. For 0 < x <π/2, tan x increases as x increases and assumes arbitrarily large positive values as x approaches to π/2. Cosec x decreases for x ∈ (3π/2, 2π) and assumes arbitrarily large negative values as x approaches to 2π. The solutions of a trigonometric equation for which 0 ≤ x < 2π are called principal solutions. The expression involving integer ‘n’ which gives all solutions of a trigonometric equation is called the general solution. If in a circle of radius r, an arc of length l subtends an angle of θ radians, then l = r θ. Key words: Relation between radian and real numbers, Domain and range of trigonometric functions, Trigonometric Functions of Sum and Difference of Two Angles

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