Trigonometric Functions | Class 11 Mathematics Notes

Domain and Range of Trigonometric Functions

This section discusses the domain and range of the six trigonometric functions.

• Sine and cosine functions are defined for all real numbers (domain: R). Their values lie between –1 and 1 (range: [–1, 1]).

• Tangent and cotangent functions are defined for all real numbers except where their denominators are zero. Specifically, tan x is undefined where cos x = 0 (x ≠ (2n+1)π/2), and cot x is undefined where sin x = 0 (x ≠ nπ), for integers n. Their ranges are all real numbers (–∞, ∞).

• Secant and cosecant functions are defined where their denominators are not zero. Sec x is undefined where cos x = 0, and cosec x is undefined where sin x = 0. Their ranges are (–∞, –1] ∪ [1, ∞).

The section also explains how the values of these functions change as the angle varies within each quadrant, illustrating the continuous and periodic nature of sine and cosine, and the discontinuities in tangent and cotangent functions.

Understanding domain and range is crucial for graphing these functions and solving equations involving them.

📊 Diagram: Table on page 11 (1×3); Table on page 11 (6×5)

🧪 Activity: Activity: Identifying domain and range of given trigonometric functions from graphs.

🔗 Connection: This section prepares for graphing trigonometric functions, discussed next.

Table on page 11 (1×3)

| {y : y ∈ R, y ≤ – 1or y ≥ 1}. The domain of y = tan x is the set {x : x ∈ R and π x ≠ (2n + 1) , n ∈ Z} and range is the set of all real numbers. The domain of 2 y = cot x is the set {x : x ∈ R and x ≠ n π, n ∈ Z} and the range is the set of all real numbers. π We further observe that in the first quadrant, as x increases from 0 to , sin x 2 π increases from 0 to 1, as x increases from to π, sin x decreases from 1 to 0. In the 2 3π third quadrant, as x increases from π to , sin x decreases from 0 to –1and finally, in 2 3π the fourth quadrant, sin x increases from –1 to 0 as x increases from to 2π. 2 Similarly, we can discuss the behaviour of other trigonometric functions. In fact, we have the following table: I quadrant II quadrant III quadrant IV quadrant sin increases from 0 to 1 decreases from 1 to 0 decreases from 0 to –1 increases from –1 to 0 cos decreases from 1 to 0 decreases from 0 to – 1 increases from –1 to 0 increases from 0 to 1 tan increases from 0 to ∞ increases from –∞to 0 increases from 0 to ∞ increases from –∞to 0 cot decreases from ∞ to 0 decreases from 0 to–∞ decreases from ∞ to 0 decreases from 0to –∞ sec increases from 1 to ∞ increases from –∞to–1 decreases from –1to–∞ decreases from ∞ to 1 cosec decreases from ∞ to 1 increases from 1 to ∞ increases from –∞to–1 decreases from–1to–∞ | | |

Table on page 11 (6×5)

Table on page 15 (1×3)

| Find the values of other five trigonometric functions in Exercises 1 to 5. 1 1. cos x = – , x lies in third quadrant. 2 3 2. sin x = , x lies in second quadrant. 5 3 3. cot x = , x lies in third quadrant. 4 13 4. sec x = , x lies in fourth quadrant. 5 5 5. tan x = – , x lies in second quadrant. 12 Find the values of the trigonometric functions in Exercises 6 to 10. 6. sin 765° 7. cosec (– 1410°) 19π 11π 8. tan 9. sin (– ) 3 3 15π 10. cot (– ) 4 3.4 Trigonometric Functions of Sum and Difference of Two Angles In this Section, we shall derive expressions for trigonometric functions of the sum and difference of two numbers (angles) and related expressions. The basic results in this connection are called trigonometric identities. We have seen that | | |