The definition of sin,cos and tan , let @ be any angle
1) place @ in stardard position
2) choose any point p, other than the origin, on the terminal side of @
3) draw a perpendicular from p to the x-axis
forming a reference right angle for @
4) let (x,y) donate the coordinate of p and let r be the distance from p to the origin.
Note: that r is a distance and hence is always positive. by the Pythagoras theorem.
then we define
sin@= y/r = opp/hyp
The point p on the terminal side of @ is chosen. the value of sin @ , cos@, tan@remains the same, in other words, the values of sin@,cos@ and tan @, are independent of the point p. To show thus, let p=(X1y1) and Q(X2Y2) be two distinct points on the terminal side of @ different than (0,0) . let R2 be the distance from Q to(0,0)
Using point p, we have sin @= y1/r1,
on@ X1/R1 tan @= Y1/R1, using@ we have sin@=y2/r2 , cos@= x2/r2 and tan@=y2/r2.
however , triangles OAP and OBQ are similar and hence ratios of corresponding sides and equal that is
y1/r1=y2/r2 and X1/r1=x2/r2 and y1/X1=y2/x2
Note: if p=(x,y) is chosen on the unit circle, then r=1, Sin@= y/r =, cos @ = x/r and the co-ordinate of p are given by p = ( cos@ sin@) .
if @ and (–@) and P2= ( x2 y2) are chosen on the unit circle and on the terminal side of @ and (–@) respectively, then x1=x2 , y1=y2 ), sin@= y1, Sin(–@) = y2, cos@=x2, and cos(–@)=x2. Hence for any angle@.
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