Graphs, Equations and the Sine and Cosine Rules

• $y = \sin x$ and $y = \cos x$: waves between $-1$ and $1$, period $360^\circ$ ($2\pi$). $\sin x$ starts at 0; $\cos x$ starts at its maximum 1.
• $y = \tan x$: period $180^\circ$, takes every real value, and has vertical asymptotes at $x = 90^\circ, 270^\circ, \ldots$
• For $y = a\sin(bx) + c$ (with $b > 0$), and the same for cosine: the amplitude is $|a|$, the period is $360^\circ / b$, the midline is $y = c$, and the values run from $c - |a|$ to $c + |a|$.

• Sine: $\sin x = k$ gives $x = \alpha$ and $x = 180^\circ - \alpha$, plus multiples of $360^\circ$.
• Cosine: $\cos x = k$ gives $x = \alpha$ and $x = 360^\circ - \alpha$, plus multiples of $360^\circ$.
• Tangent: $\tan x = k$ gives $x = \alpha$ plus multiples of $180^\circ$.
• Multiple angles: for $\sin(bx) = k$, solve for $u = bx$ over the range multiplied by $b$, then divide by $b$.
• Quadratic in sin x: put $s = \sin x$, solve the quadratic, and reject any root outside $[-1, 1]$.
• Inequality for every x: with $s = \sin x$, the expression only needs checking for $-1 \le s \le 1$.

In a triangle with sides $a, b, c$ facing angles $A, B, C$:

• Sine rule: $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$. Use it when you know a side and the angle facing it.
• Cosine rule: $c^2 = a^2 + b^2 - 2ab\cos C$. Use it with two sides and the angle between them, or with all three sides.
• Area: $\tfrac{1}{2}ab\sin C$, using two sides and the angle between them.
• An obtuse angle has a negative cosine, so the side facing it is longer than Pythagoras alone would give.