Trigonometric functions
INTRODUCTION
Trigonometry is the subject which deals with the properties of triangles. You might recall from your previous knowledge that in a right angled triangle the ratios between any two sides may be defined in terms of what are called trigonometric ratios, but the scope of trigonometry is not limited to this only. These rules or laws of trigonometry are used as tools for mathematical analysis of various problems in physics and engineering.
Trigonometric functions
A real number Θ can be interpreted as the measure of the angle constructed as follows: wrap a piece of string of length Θ units around the unit circle x2 + y2 = 1 (counterclockwise if Θ≥ 0, clockwise if Θ < 0) with initial point P(1, 0) and terminal point Q(x, y). This gives rise to the central angle with vertex O(0, 0) and sides through the points P and Q. All six trigonometric functions of Θ are defined in terms of the coordinates of the point Q(x,y), as follows:
cos Θ = x sec Θ = 1/x if x ≠ 0
sin Θ = y cosec Θ = 1/y, if y ≠ 0
tan Θ = y/x if x ≠ 0 cot Θ = x/y if y ≠ 0
Since Q(x,y) is a point on the unit circle, we know that x2 + y2 = 1. This fact and the definitions of the trigonometric functions give rise to the following fundamental identities:
Pythagorean identity sin2 Θ + cos2 Θ = 1
Reciprocal identities
sec Θ = 1/cos Θ
cosec Θ = 1/sin Θ
tan Θ = sin Θ/cos Θ
tan Θ = 1/cot Θ
cot Θ = 1/tan Θ
This modern notation for trigonometric functions is due to L. Euler (1748).
More generally, if Q(x,y) is the point where the circle x2 + y2 = R2 of radius R is intersected by the angle Θ, then it follows (from similar triangles) that
cos Θ = x/R sec Θ = R/x, if x ≠ 0
sin Θ= y/R cosec Θ = R/y, if y ≠ 0
tan Θ = y/x, if x ≠ 0 cot Θ = y/x if y ≠ 0
Periodic Functions
If an angle Θ corresponds to a point Q(x, y) on the unit circle, it is not hard to see that the angle Θ + 2Π corresponds to the same point Q(x, y), and hence that
cos (Θ + 2Π) = cos Θ, sin (Θ + 2Π) = sin Θ …(1)
Moreover, 2Π is the smallest positive angle for which Equations (1) are true for any angle Θ. In general, we have for all angles Θ:
cos (Θ + 2nΠ) = cos Θ, sin (Θ + 2nΠ) = sin Θ, n = 0, ±1, ±2 …. …(2)
We call the number 2Πthe period of the trigonometric functions sin and cos, and refer to these functions as being periodic. Both sec and cosec are periodic functions as well, with period 2Π, while tan and cot are periodic with period Π.
Illustrations
1. Find the period of the function
f(x) = – 3 cos (3x).
Sol. The function f(x) = – 3 cos (3x) runs through a full cycle when the angle 3x runs from 0 to 2Π, or equivalently when x goes from 0 to 2Π/3. The period of f(x) is then 2Π/3.
2. Find the period of the function f(t) = 8 sin (7Πt).
Sol. For the function f(t) = 8 sin (7Πt) to run through a full cycle, the angle 7Πt should run from 7Πt = 0 to 7Πt = 2Π, and hence t should run from t = 0 to t = 2/7. The period of f(t) is then 2/7.