0For a right angled triangle, with one angle θ (which is not the right angle), the trigonometric ratios are defined as follows:
sin θ = Perpendicular / Hypotenuse (corresponding to the angle θ)
cos θ = Base / Hypotenuse (corresponding to the angle θ)
tan θ = Perpendicular / Base (corresponding to the angle θ)
Notice that tanθ = sinθ/cosθ.
Three other ratios are also defined as the inverse of the basic three ratios.
cosec θ = 1/sinθ
sec θ = 1/cosθ
cot θ = 1/tanθ = cosθ/sinθ
By applying the Pythagoras theorem to the right angled triangle, we can deduce the following identities:
- sin²θ + cos²θ = 1 (use Pythagoras theorem for AB,BC and AC)
- 1 + cot²θ = cosec²θ (divide the first identity by sin²θ)
- 1 + tan²θ = sec²θ (divide the first identity by cos²θ)
From the right angled triangle, we can also see that if one angle is θ, the other acute angle is (90-θ). Taking the same three ratios for the angle (90-θ), we find that:
sin(90-θ) = cosθ,
cos(90-θ)= sinθ,
tan(90-θ)=cotθ.
