1# Preliminary topics 2 3## Circular functions 4 5{#id .class height=150px} 6{#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{#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- {#id .class width=40%} The angle at the centre of a circle is twice the angle at the circumference subtended by the arc 49- {#id .class width=40%} the angle in a semicircle is a right angle 50- {#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$$|x|=\sqrt{x^2}$$