By considering the relationships between the sides of the right angled triangle (hypotonuese of 12 cm) explain why sin x can never be greater than 1?

We know,

`sin theta = "opposite"/"hypotenuse"`

Now, we know that, In a Triangle, The side opposite to the greater angle is greater than the other.

In a Right Angled Triangle,

The Right Angle is the greatest angle.

So, The opposite side to it must be the greatest.

So, Hypotenuse is the greatest side.

That means, `"Opposite " lt " Hypotenuse"`.

So, `"Opposite"/"Hypotenuse" lt 1` [It is an proper fraction].

I.e `sin theta lt 1`.

You can prove it with the Trigonometric Identities too.

We all know, `sin^2 theta + cos^2 theta = 1`

As, `cos^2 theta` is a square term, it to be real, has to be positive.

So, `cos^2 theta gt 0`

So, `-cos^2 theta lt 0`

`rArr 1 - cos^2 theta lt 1`

`rArr sin^2 theta lt 1` [From the identity]

`rArr sin^2 theta - 1 lt 0`

`rArr (sin theta + 1)(sin theta - 1) lt 0`

Now, Either `sin theta lt -1` or `sin theta lt 1`.

We know, Sine is a Periodic Function whose value ranges from `-1` to `1`. (`-1 <= sin x <= 1`).

So, `sin theta cancellt -1`. But, `sin theta lt 1`.

Hence Proved again.

Hope this helps.

The trig functions are based on the Pythagorean Theorem

`A^2 + B^2 = C^2`

`sin = A`
`cos = B`
`hyp = C`

so

`sin^2 + cos^2 = hyp^2`

In the classic unit trig triangle the Hypothenuse is 1 so

`sin^2 + cos^2 = 1^2`

This can be illustrated by a `45` degree right triangle

`Sin = cos` so

`sin 45 = 0.707`
`cos 45 = 0.707`

`0.707^2 + 0.707^2 =1^2`

`0.5 + 0.5 = 1`

`1 =1` Classic Pythagorean theorem.

Now if the value of the sin is 1 the value of the cos must be 0

`1^2 + 0^2 = 1^2`

`1 = 1`

So if the angles of the triangle are such that `sin =1 and cos =0`there is no longer a triangle but just a vertical line because the line adjacent is 0

The value of the `Sin` cannot be more than 1 because the triangle that the trig functions are based on no longer exists.