Andrew's git - notes.git/blob

   1# Preliminary topics
   2
   3## Circular functions
   4
   5![](../methods/graphics/exact-values-1.png){#id .class height=150px}
   6![](../methods/graphics/exact-values-2.png){#id .class height=150px}
   7
   8$\sin \theta$ - $y$-coord on unit circle  
   9$\cos \theta$ - $x$-coord on unit circle  
  10$\tan \theta = {\sin \theta \over \cos \theta}$
  11
  12$1^\text{c}= {180^\circ \over \pi}  \quad \text{or} \quad 1^\circ = {\pi^\text{c} \over 180}$
  13
  14period = $2 \pi \over n$
  15
  16## Sine and cosine rules
  17
  18### Sine rule
  19
  20$${a \over \sin A}={b \over \sin B}={c \over \sin c}$$
  21
  22### Cosine rule
  23
  24$$a^2=b^2 - 2bc \cos A$$
  25
  26## Geometry
  27
  28### Area of a triangle
  29
  30$A={1 \over 2} a b \sin C$
  31
  32### Parallel lines
  33
  34If parallel lines are crossed by transversal:
  35
  36- alternate angles are equal
  37- corresponding angles are equal
  38- co-interior angles are supplementary
  39
  40![](graphics/transversal.png){#id .class width=40%}
  41
  42### Angles in a polygon
  43
  44Sum of interior angles of $n$-sided polygon is $(n-2) \times 180^\circ$
  45
  46### Circle geometry
  47
  48- ![](graphics/circle-centre-angles.png){#id .class width=40%} The angle at the centre of a circle is twice the angle at the circumference subtended by the arc
  49- ![](graphics/semicircle-right-angle.png){#id .class width=40%} the angle in a semicircle is a right angle
  50- ![](graphics/segment-angles.png){#id .class width=40%} angles in the same segment of a circle are equal
  51- ![]()
  52
  53## Circles, ellipses and hyperbolas
  54
  55Standard form is $Ax^2+By^2+Cx+Dy=0$
  56
  57- if $A+B$ then circle
  58- if $A>0$ and $B>0$ and $A\ne B$ then ellipse
  59- if $A<0<B$ or $B<0<A$ then hyperbola
  60
  61### Circles
  62
  63$$(x-h)^2 + (y-k)^2 = r^2$$
  64
  65- centre $(h,k)$
  66- radius $r$
  67
  68### Ellipses
  69
  70$${(x-h)^2 \over a^2}+{(y-k)^2 \over b^2} = 1$$
  71
  72- centre $(h, k)$
  73- $x$-radius $a$
  74- $y$-radius $b$
  75- $\therefore \text{width}=2a, \quad \text{height}=2b$
  76
  77### Hyperbolas
  78
  79$${(x-h)^2 \over a^2} - {(y-k)^2 \over b^2} = 1$$
  80
  81- centre at $(h,k)$
  82- asymptotes at $y-k=\pm{b \over a}(x-h)$
  83
  84${(x-h)^2 \over a^2} - {(y-k)^2 \over b^2} = 1$ and ${(y-k)^2 \over b^2} - {(x-h)^2 \over a^2} = 1$ are **conjugate hyperbolas**
  85
  86## Modulus function
  87
  88$$|x|=\sqrt{x^2}$$
  89
  90## Parametric equations
  91
  92### Circles
  93$$\[\begin{cases}
  94        x=a\cos t\\
  95        y=a\sin t
  96    \end{cases}
  97\text{where radius} =a$$
  98
  99To convert to cartesian, factorise and use $\cos^2 x + \sin^2 x=1$
 100
 101$\cos^2 t + \sin^2 t = 1$  
 102$\implies {\cos^2 \over \sin^2 t} + {\sin^2 t \over sin^2 t} = {1 \over \sin^2 t} \implies \csc^2 t - \cot^2 t$