Meaning of trigonometric functions

Question

When we define trigonometric ratios using a right triangle then it denotes the ratio of its side but what does a trigonometric function of angle greater than 90° denotes in a unit circle ?

Answer 1

In a unit circle: Take a point $(x,y)$ on the circle (that means $x^2 + y^2 =1$ by the way but we don't need to worry about that yet.)

Look at the angle from $(x,y)$ to the origin, to the $x$ axis. Call that angle $\theta$.

Notice, for every possible angle, $\theta$ from $0$ to $360$ (including $0$ but not including $360$) there is exactly one pair of $(x,y)$ on the circle, and for every pair of $(x,y)$ on the circle there is exactly one angle, $\theta$.

We define $\cos \theta$ to be equal to the $x$ value.

We define $\sin \theta$ to be equal to the $y$ value.

And that is it.

If $\theta < 90^{\circ}$ then the three points. $(x,y)$ and the origin and $(x, 0)$ on the $x$-axis, form a right triangle with hypotenuse $1$. Those the circle definition fits in perfectly with the right triangle definition.

If you want to use the right triangle definition for angles larger than $90$ then well you have to imagine sides of triangles of negative value (that is they go to the negative values of the $x$ axis or $y$-axis.)

Imagine we have a right triangle but instead of facing to the right it faces to the left. If the angle is $135$ or a $45$ past $90$ and we say the bottom leg has a negative distance of $-1$ and the opposite leg has a height of $1$ and the hypotenuse is $\sqrt{2}$ we get with the right triangle definition that

$\cos 135 =-\frac 1{\sqrt 2}$ and $sin 135= \frac 1{\sqrt 2}$.

Basically we have to think of these as special cases.

IMO it's much easier to just use the circle definition.

The point in a circle will also form a right triangle but you just need to keep track which quadrant of the circle it lies in and make those values positive or negative as nescessary. In that way you can use the rectangle idea.