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Trigonometric functions can be defined in two different but equivalent ways : (a) Trigonometric Functions of Real Numbers & (b) Trigonometric Functions of Angles. First we shall discuss trigonometric functions of real numbers.
THE UNIT CIRCLE
The unit circle is the circle of radius 1 and centre at the origin in the x-y plane. Its equation is x 2 + y2 = 1
1f 3f f f Example : Are the points P @ , and Q f, f on the unit circle? 2 2 2 2
f f g
w w w w w g w pff 1f f3 f f fff f ff ff
Any point on the unit circle will satisfy the equation x 2 + y 2 = 1 A Since 3f 1f f f f f = f+ f= 1 2 2 4 4 P is on the unit circle A
w g2 w w w w w g2 pff f 1f f3 f fff ff ff f f
f g2 f g2 3f 1f f f f f
1f 9f 10f f f ff f f f f = f+ f= ff≠ 1 4 4 4
Q is not on the unit circle A 1f f Example : If P f, y is on the unit circle, what is its y @ coordinate? 3
f g2 1f f f
+ y 2 = 1,
1f 8f f f f f y2 = 1 @ = , 9 9 w w w w w w w w w w w w w w p2 f f fffff ffff ffff s 8f= F 2ffff y =F 3 9
Terminal Points on the Unit Circle :
Let t be a real number. We start at the point (1,0) and mark of a distance t along the unit circle. If t is positive, we proceed in the counterclockwise direction and if t is negative, we proceed in the clockwise direction. Suppose we reach a point P(x,y) on the unit circle.
This point P(x,y) reached this way is called the terminal point associated with the real number t . The circumference of the unit circle is C = 2π 1 = 2π . So, for one full rotation the distance covered will be 2π .
Moving counterclockwise : If a point starts at (1,0) and move counterclockwise and . come back to (1,0), it moves a distance 2πb c t = 0 determines the terminal point 1,0 A
t = 2π also determines the terminal point 1,0 A
b b c c
t = π determines the terminal point @ 1,0 A b c πf f f t = ffdetermines the terminal point 0,1 2 b c 3πf ff ff f t = ff determines the terminal point 0, @ 1 A 2
Moving clockwise : t is negative here. If a point starts at (1,0) and move clockwise and come back to (1,0), it moves a distance 2πb c . t = 0 determines the terminal point 1,0 A
t = @ 2π also determines the terminal point 1,0 A
b b c c
t = @ π determines the terminal point @ 1,0 A b c πf f f t = @ ffdetermines the terminal point 0, @ 1 2 b c 3πf ff ff f t = @ ff determines the terminal point 0,1 A 2
TRIGONOMETRIC FUNCTIONS OF REAL NUMBERS
There are six trigonometric functions, which are sine, cosine, tangent, cotangent, secant and cosecant and are denoted by symbols sin, cos, tan, cot, sec and csc respectively. The trigonometric functions of the real number t are defined below : Let t be a real number and let P(x,y) be the terminal point on the unit circle associated with t . Then the trigonometric functions of the real number t are : yf ff f f sin t = y, cos t = x, tan t = ,x ≠ 0 x 1f 1f xf ff f f f f f ff f f sec t = , x ≠ 0 csc t = , y ≠ 0 cot t = , y ≠ 0 x y y These are also called trigonometric ratios A 3f 4f f f f Example : If the point P @ f, f on the unit circle corresponds to the real number t, 5 5 find the six trigonometric functions of t .
3f 4f f f f Here, x = @ f, y = f 5 5 4f f f f sin t = y = , 5
3f f f f xf @ff 3f ff fff f fff f f5 f f f f f cot t = = 4 = @ f f f f f y 4 5
3f f f cos t = x = @ , 5 sec t = 1ff 5f fff fff ff f f f =@ 3f f f f 3 @ 5
yf fff 4f ff fff f fff f ff f f f tan t = = 53 = @ , f f f f x @ 3
4f f f f f
1f 5f ff f f f f f csc t = 4 = f 4 f f f f
Let us take a ray OA with its vertex (endpoint) as O. If we rotate it about the vertex O , jj jj jj j j j j k j and come to the final position OB , the trigonometric angle is the measure of rotation or jj jj jj j j j j k j amount of rotation. The starting position of the ray or OA is called the initial side and jj jj jj j j j j k j the ending position or OB is called the terminal side.
jj jj jj j j j j k j
If the rotation of the ray from its initial position is in the counterclockwise direction, the angle is assigned a positive sign. If it has rotated in the clockwise direction, the angle is assigned a negative sign. A zero angle corresponds to zero rotation, in which case the initial and the terminal sides are coincident