## Intro and rationale Here we move on to a more flexible definition of the fundamental trigonometric functions using the unit circle. This allows us to use trigonometry to reason about negative angles and those greater than 90$\textdegree$/$\frac{\pi}{2} \text{rad}$, which could not form part of a right triangle. ## Setup Consider the unit circle centered on the origin. Take an arbitrary point on the circumference (in the first quadrant for now) and label this $C$. Let the origin be $A$ and the point at which a perpendicular line from C intersects the x-axis be $B$. Let the acute $\angle CAB = \theta$, and name each side of the resulting right triangle $\Delta ABC$ $a$,$b$,$c$ after the corresponding opposite vertex. Let the coordinates of $C$ be $(x,y)$ as per the diagram below: ![[Unit Circle with Triangle.png]] [^1] What can we say about $(x,y)$ ? Well $ \begin{align*} x &= c \; \text{...it is the opposite side of a rectangle from c}\\ y &= a \; \text{...it is the length of a}\\ b &= 1 \; \text{...by construction, radius of a unit circle}\\ \\ \cos \theta &= \frac{c}{b} = \frac{c}{1} = c \\ \sin \theta &= \frac{a}{b} = \frac{a}{1} = a \\ \\ \\ \text{therefore } (x,y) &= (\cos \theta, \sin \theta) \tag{1} \end{align*} $ This is an extremely important result for many applications. ### Projection As a piece of nomenclature, the line $c$ is sometimes referred to as the $x$-component of $b$, or the _projection_ of $b$ onto the $x$-axis. Imagine a light source high in the positive direction of the $y$-axis shining down towards $y=0$. The "shadow" cast by $b$ would be $c$. Likewise $a$ is the $y$-component of $b$ or the projection of $b$ onto the $y$-axis. ## Proof of the Pythagorean Trigonometric Identity via the unit circle definition Given all the above, note that: $ \begin{align*} b^2 &= a^2 + c^2 &\text{...by Pythagoras} \\ \text{so} \; a^2 + c^2 &= 1 & \text{...since b=1} \\ \text{but} \sin \theta &= a \text{ and } \cos \theta = c \\ \\ \text{therefore } \sin^2 \theta + \cos^2 \theta &= 1 \\ \end{align*} $ ...which is somewhat more straightforward than the previous [[Pythagorean Trigonometric Identities#Proof that $ sin 2 theta + cos 2 theta = 1$|proof]] using the right triangle definition. Having established the fundamental identity via this alternative method, the derivation of the other two pythagorean identities is the same as before. ## Tangents in terms of the unit circle We know algebraically that $\tan \theta = \frac{\sin \theta}{\cos \theta}$ ... but how do we understand this here? Or, to put that another way, what is the geometric formulation of $\tan$ that applies to the unit circle definition above? If we look at the difference between the origin and point C, we can see that $\sin \theta$ is the change in $y$ and $\cos \theta$ is the change in $x$ so $\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\Delta y}{\Delta x} = \frac{rise}{run}$ ...and therefore see that geometrically, $\tan \theta$ is the slope of $AC$. This will be very useful for understanding the [[Geometric Construction of Trigonometric Identities from the Unit Circle|identities]] which arise. See [Khan Academy Review Article](https://www.khanacademy.org/math/trigonometry/unit-circle-trig-func/unit-circle-definition-of-trig-functions/a/trig-unit-circle-review) [[Radians]] [^1]: Diagram available on [Geogebra](https://www.geogebra.org/calculator/btz38m3c). Feel free to use it if it's helpful.