[spec] reciprocal differentiation

diff --git a/spec/prelim.md b/spec/prelim.md
index ed06b09db8ad416577a72c0d21ee35889b9363de..d158bbb0eb31300337d3c7939de5c6e15b0bca0c 100644 (file)
--- a/spec/prelim.md
+++ b/spec/prelim.md
@@ -2,6 +2,9 @@
 
 ## Circular functions
 
+![](../methods/graphics/exact-values-1.png){#id .class height=150px}
+![](../methods/graphics/exact-values-2.png){#id .class height=150px}
+
 $\sin \theta$ - $y$-coord on unit circle  
 $\cos \theta$ - $x$-coord on unit circle  
 $\tan \theta = {\sin \theta \over \cos \theta}$
@@ -47,14 +50,31 @@ Sum of interior angles of $n$-sided polygon is $(n-2) \times 180^\circ$
 - ![](graphics/segment-angles.png){#id .class width=40%} angles in the same segment of a circle are equal
 - ![]()
 
+## Circles, ellipses and hyperbolas
+
+Standard form is $Ax^2+By^2+Cx+Dy=0$
+
+- if $A+B$ then circle
+- if $A>0$ and $B>0$ and $A\ne B$ then ellipse
+- if $A<0<B$ or $B<0<A$ then hyperbola
+
+### Circles
+
+$$(x-h)^2 + (y-k)^2 = r^2$$
 
-## Ellipses and hyperbolas
+- centre $(h,k)$
+- radius $r$
 
-#### Ellipses
+### Ellipses
 
 $${(x-h)^2 \over a^2}+{(y-k)^2 \over b^2} = 1$$
 
-#### Hyperbolas
+- centre $(h, k)$
+- $x$-radius $a$
+- $y$-radius $b$
+- $\therefore \text{width}=2a, \quad \text{height}=2b$
+
+### Hyperbolas
 
 $${(x-h)^2 \over a^2} - {(y-k)^2 \over b^2} = 1$$
 
@@ -65,4 +85,18 @@ ${(x-h)^2 \over a^2} - {(y-k)^2 \over b^2} = 1$ and ${(y-k)^2 \over b^2} - {(x-h
 
 ## Modulus function
 
-$$|x|=\sqrt{x^2}$$
\ No newline at end of file
+$$|x|=\sqrt{x^2}$$
+
+## Parametric equations
+
+### Circles
+$$\[\begin{cases}
+        x=a\cos t\\
+        y=a\sin t
+    \end{cases}
+\text{where radius} =a$$
+
+To convert to cartesian, factorise and use $\cos^2 x + \sin^2 x=1$
+
+$\cos^2 t + \sin^2 t = 1$  
+$\implies {\cos^2 \over \sin^2 t} + {\sin^2 t \over sin^2 t} = {1 \over \sin^2 t} \implies \csc^2 t - \cot^2 t$