vault backup: 2024-10-02 11:37:51

.obsidian/file-recovery.json

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education/math/MATH1060 (trig)/Graphing.md

@@ -74,11 +74,53 @@ Given the form $y = A\tan(Bx - C) + D$ (the same applies for $\cot$)
 - The range of both is $(-\infty, \infty)$
 - The phase shift is $\frac{C}{B}$
 - The vertical shift is $D$
+
+# Secant
+$$ y = \sec(x) $$
+![Graph of secant](assets/graphsec.jpg)
+
+$$ sec(x) = \frac{1}{\cos{x}} $$
+Because secant is the reciprocal of cosine, when $\cos{x} = 0$, then secant is undefined. $|\cos$| is never *greater than* 1, so secant is never *less than* 1 in absolute value. When the graph of cosine crosses the x axis, an asymptote for a matching graph of secant will appear there. 
+
+The general form of secant is:
+$$ y = A\sec(B{x} - C) + D $$
+$A$, $B$, $C$, and $D$ will have similar meanings to the secant function as they did to the sine and cosine functions. 
+
+# Cosecant
+$$ y = \csc(x) $$
+![Graph of cosecant](assets/graphsec.jpg)
+
+$$ \csc(x) = \frac{1}{\sin(x)} $$
+
+Because cosecant is the reciprocal of sine, when $\sin{x} = 0$, then cosecant is undefined. $|\sin$| is never *greater than* 1, so secant is never *less than* 1 in absolute value. When the graph of sine crosses the x axis, an asymptote for a matching graph of cosecant will appear there. 
+
+The general form of cosecant is:
+$$ y = A\csc(B{x} - C) + D $$
+$A$, $B$, $C$, and $D$ will have similar meanings to the cosecant function as they did to the sine and cosine functions. 
+
+# Features of Secant and Cosecant
+- The stretching factor is $|A|$
+- The period is $\frac{2\pi}{|B|}$
+- The domain of secant is all $x$, where $x \ne \frac{C}{B} + \frac{\pi}{2} + \frac{\pi}{|B|}k$, where $k$ is an integer. (Every half period + phase shift is where asymptotes appear)
+- The domain of cosecant is all $x$, where $x \ne \frac{C}{B} + \frac{\pi}{|B|}k$, where $k$ is an integer.
+- The range is $(\infty, -|A| +D]\cup [|A| + D], \infty)$
+- The vertical asymptotes of secant occur at $x = \frac{C}{B} + \frac{\pi}{2} + \frac{\pi}{2} + \frac{\pi}{|B|}k$, where $k$ is an integer.
+- The vertical asymptotes of cosecant occur at $x = \frac{C}{B} + \frac{\pi}{|B|}k$, where $k$ is an integer.
+- The vertical shift is $D$. 
 # Examples
 > Given $-2\tan(\pi*x + \pi) - 1$
 
  $A = -2$, $B = \pi$, $C = -\pi$, $D = -1$
 
+> Identify the vertical stretch/compress factor, period, phase shift, and vertical shift of the function $y = 4\sec(\frac{\pi}{3}x - \frac{\pi}{2}) + 1$ 
+
+$A = 4$, $B = \frac{\pi}{3}$, $C = \frac{\pi}{2}$, $D = 4$
+
+Vertical stretch: $|4| = 4$
+Period: $\frac{2\pi}{\frac{\pi}{3}} = \frac{2\pi}{1} * \frac{3}{\pi} = 6$
+Phase shift: $\dfrac{\frac{\pi}{2}}{\frac{\pi}{3}} = \frac{3}{2}$
+Vertical shift: $1$
+
 | Transformation | Equation                  |
 | -------------- | ------------------------- |
 | Stretch        | $\|-2\| = 2$              |