features of asymptotes on tan graphs

diff --git a/methods/circ-functions.md b/methods/circ-functions.md
index a48b106a03b2820f6413371572302a9c9bc26859..f30ed7adc4874089ba307799e81c327921756838 100644 (file)
--- a/methods/circ-functions.md
+++ b/methods/circ-functions.md
@@ -6,6 +6,8 @@ $$1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}$$
 
 ## Exact values
 
+
+
 ## $\sin$ and $\cos$ graphs
 
 $$f(x)=a \sin(bx-c)+d$$
@@ -22,6 +24,19 @@ Range is $[-b+c, b+c]$;
 
 Graph of $\cos(x)$ starts at $(0,1)$. Graph of $\sin(x)$ starts at $(0,0)$.
 
+**Mean / equilibrium:** line that the graph oscillates around ($y=d$)
+
+## Solving trig equations
+
+1. Solve domain for $n\theta$
+2. Find solutions for $n\theta$
+3. Divide solutions by $n$
+
+$\sin2\theta={\sqrt{3}\over2}, \quad \theta \in[0, 2\pi] \quad(\therefore 2\theta \in [0,4\pi])$
+$2\theta=\sin^{-1}{\sqrt{3} \over 2}$
+$2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}$
+$\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}$
+
 ### Amplitude
 
 Amplitude of $a$ means graph oscillates between $+a$ and $-a$ in $y$-axis
@@ -70,4 +85,5 @@ $n$ is $y$-dilation ($\equiv$ amplitude)
 period $T$ is $\pi \over n$
 range is $R$
 roots at $x={k\pi \over n}$
-asymptotes at $x={{(2k+1)\pi}\over 2},\quad k \in \mathbb{Z}$
+asymptotes at $x={{(2k+1)\pi}\over 2n},\quad k \in \mathbb{Z}$
+**Asymptotes should always have equations and arrow pointing up**