Fundamental relations between the trigonometric ratios of an angle:
To know the relations between the trigonometric ratios from the above figure, we see;
sin θ = perpendicular/hypotenuse = MP/PO and
cosec θ = hypotenuse/perpendicular = PO/MP
It is clear that one is the reciprocal of the other.
So, sin θ = 1/cosec θ and
cosec θ = 1/sin θ ………. (a)
Again, cos θ = base/hypotenuse = OM/OP and
sec θ = hypotenuse/ base = OP/OM
One is reciprocal of the other.
That is, cos θ = 1/sec θ and sec θ = 1/cos θ ………. (b)
So, tan θ = perpendicular/base = MP/OM and cot θ = base/perpendicular = OM/MP
tan θ = 1/cot θ and cot θ = 1/tan θ ………. (c)
Moreover, sin θ/cos θ = (MP/OP) ÷ (OM/OP) = (MP/OP) × (OP/OM) = MP/OM = tan θ
Therefore, sin θ/cos θ = tan θ ………. (d)
and cos θ/sin θ = (OM/OP) ÷ (MP/OP) = (OM/OP) × (OP/MP) = OM/MP = cot θ
Therefore, cos θ/sin θ = cot θ ………. (e)
Sin θ = PM/OPThis is how the ratios are related to show that one is the reciprocal of the other according to the relations between the trigonometric ratios.
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