Today is our topic of discussion Definition of ratios .
Definition of ratios
Take a point P(x, y) on a rotating ray OZ of standard position of any angle θ where OP = r(>0). then in angle θ
sine ratio, sinθ= y/x
cosine ratio, cosθ= x/2
tangent ratio, tanθ= y/x [When x ≠0|
cotθ= x/y
cotangent ratio, [When y ≠0]
secant ratio, sec theta= r x [ When x ne0]
cosecant ratio, cosec = [When y ≠ y
It is to be observed that, P(x,y) , P’ * (x’, y’) are two point on the ray OZ where OP =r(>0) , z OP’ = r'(> 0); x, x’ and y, y’ have same signs. So from ΔΟΡΜ and triangle O * P’ * M’ x/r = (x’)/(r’), y/r = (y’)/(r’) etc.
So Tthe ratios of the angle @ do not depend on the position of the point P on the ray OZ.
If is an acute angle, then in the right angled triangle OPM hypotenuse OP = r , adjoining
side OM = x opposite side PM = y Therefore, sin theta = y/r = opposite side
P(x,y)
T
y
hypotenuse cos theta = x/r = (a * djoiningsi * de)/(hypotenuse)
tan theta = y/x = (oppositesi * de)/(a * djoiningsi * de) etc
Therefore, in case of acute angle the definition of trigonometric ratio based on coordinate and in class 9 and 10 definition based on right angled triangle are same.
Ratios of the angles 0° and 90°:
The rotating ray lies on the segment OX in case of 0 deg Therefore, P(x,0) and r = OP = x So,
sin 0 deg = y/r = 0/r = 0
cos 0 deg = x/r = x/x = 1
The rotating ray lies on the segment OY in case of 90 deg . Therefore, P(0,y) and r = OP = y
sin 90 deg = y/r = y/y = 1 cos 90 deg = x/r = 0/r = 0
From the definition it is seen that, for any angle the following rules are applicable for trigonometric ratios.
1. sin^2 theta + cos^2 theta = 1
Proof: sin theta = y/r cose = r ^ 2 = x ^ 2 + y ^ 2
sin^2 theta + cos^2 theta = (x ^ 2 + y ^ 2)/(r ^ 2) = (r ^ 2)/(r ^ 2) = 1
2. tan theta = (sin theta)/(cos theta) sec(theta) = 1/(cos theta) cot theta = (cos theta)/(sin theta) co * sec(theta) = 1/(sin theta) where the ratios are defined.
3. Considering the signs of different coordinates in the above figure,
4. sine≤ 1, cos@| ≤ 1
Proof: sin^2 theta + cos^2 theta = 1 sin^2 theta <= 1, cos^2 theta <= 1 i .e.,|sin theta|<=1,|cos theta|<=1
5. The values of sine, cose and tane for different values of are following:
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