Sin, cos and tan | Mathematico

The unit circle

So far, we have only considered the domain of $\sin, \cos$ and $\tan$ to be

0^{\circ} < θ < 90^{\circ}

but the definition of these functions can be extended by considering a point $(x, y)$ on the circle

x^{2} + y^{2} = 1

Notice that

\sin (θ) = \frac{opposite}{hypotenuse} = \frac{y}{1} = y

and

\cos (θ) = \frac{adjacent}{hypotenuse} = \frac{x}{1} = x

so that the point on the circle has coordinates

(\cos (θ), \sin (θ))

By looking at the coordinates of $(x, y)$ we can work out $\sin$ and $\cos$ for any value of $θ$ .

Additionally, we have

\tan (θ) = \frac{y}{x} = \frac{\sin (θ)}{\cos (θ)}

We'll touch more upon this fact later.

If we consider the $y$ coordinate of the point $P,$ we find the value of $\sin (θ) .$ If we plot the curve of $\sin (θ)$ we would get the following

It is worth noticing that

- 1 \leq \sin θ \leq 1

We can do similar for $\cos (θ) .$ To get the graph of $\cos (θ),$ we consider the $x$ coordinate of the point $P$ on the unit circle.

Similar to $\sin,$ we have

- 1 \leq \cos θ \leq 1

Finally, if we calculate $\frac{\sin (θ)}{\cos (θ)}$ we get $\tan (θ)$

Solve the equations