Question

How do you find the exact value of sin (0)?

Answer 1

Hint: By definition, in a right−angled triangle with length of the side opposite to angle θ as perpendicular (P), base (B) and hypotenuse (H):
sin θ = $ \dfrac{P}{H} $ , cos θ = $ \dfrac{B}{H} $ , tan θ = $ \dfrac{P}{B} $ .
Recall that the Pythagoras' theorem holds true for every right−angled triangle: $ {{P}^{2}} $ + $ {{B}^{2}} $ = $ {{H}^{2}} $ . If we assume one of the non-right-angles as θ and write an expression in terms of P, B and H for sin θ using the definition above, we will get the result. What happens to the values of P, B and H when θ = 0?

Complete step-by-step answer:
Let's say we have a right-angled triangle with the side opposite to the angle θ as P (perpendicular) and H as the hypotenuse. The third side, adjacent to the angle, calls it B (base).
It can be represented as follows:

If the angle θ = 0 and if we maintain the right-angle (trigonometric ratios are defined for right angled triangles only), then the length of P will become 0 and H and B will coincide with each other, i.e. H = B.
We know that, because of the Pythagoras' theorem, $ {{P}^{2}} $ + $ {{B}^{2}} $ = $ {{H}^{2}} $ . In this case also, $ {{0}^{2}} $ + $ {{B}^{2}} $ = $ {{H}^{2}} $ , which is true, because B = H (since angle θ = 0).
By definition, sin θ = $ \dfrac{Perpendicular}{Hypotenuse} $ = $ \dfrac{P}{H} $ . Writing θ = 0 and substituting and P = 0, we get:
⇒ sin 0 = $ \dfrac{0}{H} $
⇒ sin 0 = 0

Note: It can also be seen that cos 0 = $ \dfrac{B}{H} $ = 1, because B = H.
There are many ways to prove the Pythagoras' theorem. For instance, it can be proved by using the properties of similar triangles, by drawing a perpendicular on the hypotenuse from the right-angled vertex and observing that the two smaller triangles have the same values of the angles and are thus similar.
It can also be observed from the right angled-triangle, that sin ( $ 90{}^\circ $ − θ) = cos θ etc.
Using the Pythagoras' theorem, we can also show that $ {{\sin }^{2}}\theta $ + $ {{\cos }^{2}}\theta $ = 1.