Trigonometry triangle Question

[basty]

For the triangle shown in the below image,

$\sin 60° = \frac{opposite}{hypotenuse} = \frac{y}{r}$
$\sin 30° = \frac{opposite}{hypotenuse} = \frac{x}{r}$

The questions are:

1. What is the opposite and hypotenuse of sin 90°?
2. I am guessing that the opposite and hypotenuse of sin 90 is
r and y respectively so that $\sin 90° = \frac{opposite}{hypotenuse} = \frac{r}{y}$.
Why sin 90° = 1?

 

[jedishrfu]

If you use the opposite over the hypotenuse definition then you'd get sin(90) = r/r = 1

 

[basty]

If you use the opposite over the hypotenuse definition then you'd get sin(90) = r/r = 1

Do you mean that the sin 90° is: the opposite = r and the hypotenuse is = r too?

 

[jedishrfu]

yes. Go back and look at the definition of the hypotenuse its the side opposite the right angle right?

https://en.wikipedia.org/wiki/Hypotenuse

 

[basty]

How do you obtain $\cos 90° = \frac{adjacent}{hypotenuse} = \frac{x}{r} = 0$?

 

[SteamKing]

How do you obtain $\cos 90° = \frac{adjacent}{hypotenuse} = \frac{x}{r} = 0$?

Look at your triangle. As the angle θ gets larger, the length of the base x gets smaller. The radius r remains constant. When θ = 90°, what must x be?

 

[basty]

Look at your triangle. As the angle θ gets larger, the length of the base x gets smaller. The radius r remains constant. When θ = 90°, what must x be?

There is no θ in my triangle. There are 90°, 60°, and 30° angles. Which one you meant as the θ?

 

[SteamKing]

There is no θ in my triangle. There are 90°, 60°, and 30° angles. Which one you meant as the θ?

The angle which is labeled 60° in your diagram.

The trig functions are defined as in this diagram:

 

[basty]

Look at your triangle. As the angle θ gets larger, the length of the base x gets smaller. The radius r remains constant. When θ = 90°, what must x be?

Which side you meant by the radius?

The θ you meant is the one labeled as 60°, but the cos 90° I meant is the right angle.

What should the cos 90° be if the angle is the right angle, not the 60°?

 

[SteamKing]

Which side you meant by the radius?

In your diagram of the triangle, the side labeled "r", which I indicated as "the radius r".

The θ you meant is the one labeled as 60°, but the cos 90° I meant is the right angle.

Yes, but in order to calculate sine and cosine for 90°, you pretend to let the angle labeled 60° in your diagram increase to 90°, and examine what happens to the sides x and y in order for this to happen.

What should the cos 90° be if the angle is the right angle, not the 60°?

cos (90°) = 0, always.

Instead of using just a triangle to define the trig functions, often a unit circle (radius = 1) is used, like this:

As the point P moves counterclockwise around the circle, the coordinates x and y of P are also the values of cos (θ) and sin (θ), respectively.

Here is another diagram showing how sin (θ) varies for different angles θ drawn on a unit circle:

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