Unit Circle Introduction
The unit circle is the circle with center (0, 0) and radius 1 unit. Consider a circle with center O (0, 0) and radius 1 unit.
Using distance formula, we know that:
Hence, the equation of the unit circle is given by:
x2 + y2 = 1
Trigonometric Functions
Consider a unit circle with center at origin of the coordinate axes. Let P (a, b) be any point on the circle with angle AOP = x radian, which means that length of arc AP = x
We define Cos (x) = a and Sin (x) = b
Since ∆OMP is a right triangle, we have
OM2 + MP2 = OP2
a2 + b2 = 1 or Cos2x + Sin2x = 1
Since one complete revolution subtends an angle of 2π radian at the center of the circle, ∠ AOB = π/2, ∠ AOC = π and ∠ AOD = 3π/2. All angles which are multiples of π/2 are called quadrantal angles. The coordinates of the points A, B, C, D are respectively (1, 0), (0, 1) (-1, 0) and (0, -1). Therefore, for quadrantal angles, we have
Sin (0) = 0
Cos (π/2) = 0
Sin (π/2) = 1
Cos (π) = -1
Sin (π) = 0
Cos (3π/2) = 0
Sin (3π/2) = -1
Cos (2π) = 1
Sin (2π) = 0
Now, if we take one complete revolution from the point P, we again come back to same point P. Thus, we also observe that if x increases (decreases) by any integral multiple of 2π, the values of sine and cosine functions do not change. So we can say that Sin (2nπ+x) = Sin(x), n Є Z and Cos (2nπ+x) = Cos(x), n Є Z
Also Sin (x) = 0 implies x = nπ, where n is any integer.
Cos (x) = 0 implies x = (2n+1) π/2, where n is any integer.
Example: Find the value of Sin (31π/3)
Solution: Sin (31π/3) =
= Sin (π/3)
= √3/2
Sign of trigonometric functions
Let P (a, b) be a point of the unit circle with center at the origin such that ∠ AOP = x. If ∠ AOQ = -x, then the coordinates of the point Q will be (a, -b).
Therefore, Cos (-x) = Cos (x) and
Sin (-x) = -Sin (x)
Since for every point P (a, b) on the unit circle, -1≤a≤1 and -1≤b≤1, we have -1 ≤ Cos (x) ≤1 and -1 ≤ Sin (x) ≤ 1 for all x.
The sign of different trigonometric functions is given below:
To make the calculations using signs of different trigonometric functions easy memorize the quotation “All Silver Tea Cups” which means that ‘All’ trigonometric functions are positive in 1st quadrant, ’S’ of silver shows that sine and cosecant are positive in 2nd quadrant, ‘T’ of tea shows that tangent and cot are positive in 3rd quadrant and ‘C’ of cups shows that cosine and sec are positive in 4th quadrant.
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