In the right angled ΔXYZ, ∠XYZ = 90⁰ and a,b,c are the lengths of the sides as shown in the figure. Write the following ratios,

(i) sin X (ii) tan Z

(iii) cos X (iv) tan X.

For any right-angled triangle,

sinθ = Opposite side Side/Hypotenuse

cosθ = Adjacent sideSide/Hypotenuse

tanθ = sinθ/cosθ

= Opposite side Side/Adjacent sideSide

cotθ = 1/tanθ

= Adjacent sideSide/Opposite side Side

secθ = 1/cosθ

= Hypotenuse/Adjacent sideSide

cosecθ = 1/sinθ

= Hypotenuse/Opposite side Side

In the given triangle let us understand, the Opposite side and Adjacent sidesides.

So for ∠ X,

Opposite side Side = YZ = a

Adjacent sideSide = XY = b

So for ∠ Z,

Opposite side Side = XY = b

Adjacent sideSide = YZ = a

In general for the side Opposite side to the 90° angle is the hypotenuse.

So for Δ XYZ, hypotenuse = XZ = c

(i) sin X = Opposite side Side/Hypotenuse

= YZ/XZ

= a/c

(ii) tan Z = sinθ/cosθ

= Opposite side Side/Adjacent sideSide

= XY/YZ

= b/a

(iii) cos X= Adjacent side Side/Hypotenuse

= XY/XZ

= b/c

(iv) tan X = sinθ/cosθ

= Opposite side Side/Adjacent sideSide

= YZ/XY

= a/b