![[TrigUnitCircleFunctions.gif|500]] ![[Basic Trig Functions Graphic.png|150]] # Trigonometry ## The Unit Circle > See also: > - [[Angles]] > [!NOTE]- The Unit Circle > > ![[Pasted image 20220520163124.png|425]] > [Visualization of Trig Functions on Unit Circle](https://www.geogebra.org/m/keqhdkaj) ### Angles Greater Than 90° ## Trigonometric Functions - [ ] What do trigonometric functions represent? - [ ] Why do we use the unit circle? - [ ] When is it beneficial to treat the trig functions as ratios rather than their unit circle definitions? ### Sine & Cosine ![[Trigonometric Functions.png|475]] **From this, we can make several conclusions:** The point $(x,y)$ is equal to $(\cos \theta, \sin \theta)$ Using the pythagorean theorem, we see that: $sin^2(\theta)+cos^2(\theta) = 1^2$ ### Tangent & Secant It’s important to recognize the difference between the [[Tangent and Secant Lines]] and their corresponding trigonometric functions. ![[Tan Graphs.gif|325]] ### Cotangent & Cosecant ## Trigonometric Ratios *Specifically in right-triangles*, the trig functions may correspond to the ratios of the side lengths present. Because we’re working with triangles, there are only three “options” throughout these various ratios: **opposite**, **adjacent**, and **hypotenuse**. using basic properties of division/fractions, one value can be cancelled out when any one of these functions are divided by another (in a different one of the three sets) $sinA=\frac a c = \frac {opposite} {hypotenuse}$ $cscA=\frac c a = \frac {hypotenuse} {opposite}$ --- $cos A=\frac b c = \frac {adjacent} {hypotenuse}$ $secA=\frac c b = \frac {hypotenuse} {adjacent}$ --- $tanA=\frac a b = \frac {opposite} {adjacent}$ $cotA=\frac b a = \frac {adjacent} {opposite}$ > The phrase "SOH - CAH - TOA" is a useful mnemonic device for remembering the three main trigonometric functions. > [!attention] When Can We Use These Ratios? > > Remember that a [[Triangles#Right Triangles|hypotenuse]] isn’t just any random side, it’s specifically the side opposite to the right angle of a triangle. ### Scale Invariance of the Trigonometric Functions ## Properties of Trigonometric Functions ### Period, Amplitude, and Frequency https://en.wikibooks.org/wiki/Trigonometry/Phase_and_Frequency ### Periodic Properties ![[Pasted image 20220521034030.png|300]] ## Trigonometric Identities ### Pythagorean Identities ![[Pasted image 20220521211212.png|400]] $1+cot^2(\theta)=csc^2(\theta)$ $1+tan^2(\theta)=sec^2(\theta)$ ### Even-Odd Identities $tan(-\theta)=-tan(\theta)$ $cot(-\theta)=-tan(\theta)$ $sin(-\theta)=-sin(\theta)$ $csc(-\theta)=-csc(\theta)$ $cos(-\theta)=-cos(\theta)$ $sec(-\theta)=-sec(\theta)$