Simulation · Maths · Class 11
The six graphs and their periods
From the lesson Domain, range and graphs in Trigonometric Functions. Change the values and watch what happens.
The six graphs and their periodsMaths · Class 11
The idea behind it
NCERT §3.3.2
- sin x and cos x are defined for every real x, and their range is [−1, 1].
- tan x has domain R − {(2n + 1)π/2 : n ∈ Z} and range R; cot x has domain R − {nπ : n ∈ Z} and range R.
- sec x has domain R − {(2n + 1)π/2 : n ∈ Z}, and cosec x has domain R − {nπ : n ∈ Z}; both have range (−∞, −1] ∪ [1, ∞), because |sec x| = 1/|cos x| ≥ 1 and |cosec x| = 1/|sin x| ≥ 1.
- As x runs through the four quadrants from 0 to 2π, sin x rises from 0 to 1, falls from 1 to 0, falls from 0 to −1, then rises from −1 back to 0. cos x falls from 1 to 0, falls from 0 to −1, rises from −1 to 0, then rises from 0 to 1.
- tan x increases throughout each quadrant: from 0 towards ∞ in the first, from −∞ to 0 in the second, from 0 towards ∞ in the third and from −∞ to 0 in the fourth. Here ∞ only describes a value that grows without bound as x nears an odd multiple of π/2.
- sin, cos, cosec and sec repeat their values after every interval of 2π; tan and cot repeat after every interval of π, because tan (π + x) = tan x.
- The graph of sin x is a smooth wave between −1 and 1 crossing the x-axis at every nπ; the graph of cos x is the same wave shifted, with its peaks at 2nπ. The graph of tan x is a chain of rising branches, each between two vertical lines x = (2n + 1)π/2 that it approaches but never meets.
- Periodicity reduces any angle: cos (−1020°) = cos (−1020° + 3 × 360°) = cos 60° = 1/2, and tan (25π/4) = tan (6π + π/4) = tan (π/4) = 1.