Advanced Trigonometry Notes

Properties of the Triangle

Fundamental formulae connect the elements of a triangle, often adapted for logarithmic work through the use of a subsidiary angle. For instance, to find the side $a$ from given values of $b$, $c$, and $A$, the formula $a^2 = b^2 + c^2 - 2bc \text{cos}(A)$ can be modified. By expressing it as $a^2 = (b+c)^2 - 2bc (1 + \text{cos}(A)) = (b+c)^2 - 4bc \text{cos}^2(\frac{A}{2})$, and setting $\text{cos}^2(\theta) = \frac{4bc \text{cos}^2(\frac{A}{2})}{(b+c)^2}$, one obtains $a = (b+c) \text{sin}(\theta)$. Here, the angle $\theta$ is the subsidiary angle, determined by \text{cos}(\theta) = \frac{2\text{\sqrt{bc}} \text{cos}(\frac{A}{2})}{b+c}.

The circumcentre $O$ is the centre of the circle passing through the vertices $A$, $B$, and $C$, found by the intersection of the perpendicular bisectors of the sides. Its radius $R$ is given by R = \frac{a}{2 \text{sin}(A)} = \frac{b}{2 \text{sin}(B)} = \frac{c}{2 \text{sin}(C)} = \frac{abc}{4\text{\Delta}}, where \text{\Delta} is the area of the triangle. The in-centre $I$ and the e-centres $I_1, I_2, I_3$ are located at the intersections of internal and external angle bisectors. The in-radius $r$ is \frac{\text{\Delta}}{s}, or $r = 4R \text{sin}(\frac{A}{2}) \text{sin}(\frac{B}{2}) \text{sin}(\frac{C}{2}) = (s-a) \text{tan}(\frac{A}{2})$. The radii of the escribed circles are r_1 = \frac{\text{\Delta}}{s-a} = 4R \text{sin}(\frac{A}{2}) \text{cos}(\frac{B}{2}) \text{cos}(\frac{C}{2}) = s \text{tan}(\frac{A}{2}), and so forth.

The orthocentre $H$ is the meeting point of the perpendiculars from the vertices to the opposite sides. The pedal triangle is the triangle $DEF$ formed by the feet of these altitudes. For an acute-angled triangle, $EF = a \text{cos}(A)$ and $\angle EDF = 180^{\circ} - 2A$. The orthocentre $H$ serves as the in-centre of the pedal triangle. The distance from a vertex to the orthocentre is $AH = 2R \text{cos}(A)$, while the side lengths of the pedal triangle segments include $DE = 2R \text{cos}(B) \text{cos}(C)$.

The nine-point circle passes through the mid-points of the sides, the feet of the altitudes, and the mid-points of the segments joining the orthocentre to the vertices. Its centre $N$ is the mid-point of the segment joining the circumcentre $O$ to the orthocentre $H$. The radius of the nine-point circle is $\frac{R}{2}$. Feuerbach's Theorem states that the nine-point circle touches the in-circle and the escribed circles, with the distance between the centres given by $IN = \frac{R}{2} - r$.

The polar circle of a triangle exists when the triangle is obtuse-angled, specifically when the triangle is self-polar with respect to this circle. Its centre is the orthocentre $H$, and its radius $\rho$ satisfies $\rho^2 = -4R^2 \text{cos}(A) \text{cos}(B) \text{cos}(C)$. In an acute-angled triangle, no such circle exists in real geometry.

For any line drawn through vertex $A$ cutting the side $BC$ at $K$, let $K$ be the centroid of masses $y$ and $z$ at $B$ and $C$ respectively. With \angle BAK = \text{\beta}, \angle KAC = \text{\gamma}, and \angle AKC = \text{\theta}, the relation (y+z) \text{cot}(\text{\theta}) = y \text{cot}(B) - z \text{cot}(C) holds. Additionally, the centroid $G$ of a system of particles satisfies the relation \sum (k \times \text{\mathbf{OP}}^2) = (\sum k) \text{\mathbf{OG}}^2 + \sum (k \times \text{\mathbf{GP}}^2) for any point $O$. The medians are concurrent at the centroid $G$, which lies on the Euler line such that $OG = \frac{1}{3} OH$. The length of a median $AX$ is given by $b^2+c^2 = 2AX^2 + \frac{1}{2} a^2$.

Distances between special points are calculated using standard relations. The square of the distance between the circumcentre and in-centre is $OI^2 = R^2 - 2Rr$, and between the circumcentre and orthocentre is $OH^2 = R^2 - 8R^2 \text{cos}(A) \text{cos}(B) \text{cos}(C) = R^2 + 2\rho^2$. The distance between the orthocentre and in-centre is given by $HI^2 = 2r^2 - 4R^2 \text{cos}(A) \text{cos}(B) \text{cos}(C) = 2r^2 + \rho^2$. Similar expressions exist for the e-centres, such as $OI_1^2 = R^2 + 2Rr_1$ and $HI_1^2 = 2r_1^2 + \rho^2$.

Properties of the Quadrilateral

A convex quadrilateral $ABCD$ with sides $a, b, c, d$ and diagonals $x, y$ is cyclic if its vertices lie on a circle. The area $S$ is given by the formula S = \text{\sqrt{(s-a)(s-b)(s-c)(s-d)}}, where $s$ is the semi-perimeter. This formula, first given by the Hindu mathematician Brahmagupta, is a special case of the general quadrilateral formula. Ptolemy's Theorem for a cyclic quadrilateral states that $xy = ac + bd$, while the lengths of the diagonals are $x^2 = \frac{(ac+bd)(ad+bc)}{ab+cd}$ and $y^2 = \frac{(ac+bd)(ab+cd)}{ad+bc}$. The circumradius $R$ satisfies 4RS = \text{\sqrt{(ab+cd)(ac+bd)(ad+bc)}}. Angles can be found using $\text{tan}^2(\frac{B}{2}) = \frac{(s-a)(s-b)}{(s-c)(s-d)}$.

For a general quadrilateral, the area is $S^2 = (s-a)(s-b)(s-c)(s-d) - abcd \text{cos}^2(\frac{B+D}{2})$. If the sides are fixed, the area is maximised when the quadrilateral is cyclic, as $\text{cos}(\frac{B+D}{2}) = 0$. An extension of Ptolemy's Theorem for general quadrilaterals gives the result $x^2 y^2 = a^2 c^2 + b^2 d^2 - 2abcd \text{cos}(B+D)$. If \text{\theta} is the angle between the diagonals, the area can also be expressed as S = \frac{1}{4}(b^2 + d^2 - a^2 - c^2) \text{tan}(\text{\theta}).

A circumscribable quadrilateral is one in which a circle can be inscribed to touch all four sides. The necessary and sufficient condition for this is $a+c = b+d = s$. The area of such a quadrilateral is S = \text{\sqrt{abcd}} \text{sin}(\frac{B+D}{2}). If the quadrilateral is both cyclic and circumscribable, its area simplifies to S = \text{\sqrt{abcd}} and the in-radius is $r = \frac{S}{s}$.

Equations and Sub-Multiple Angles

General solutions to trigonometric equations are typically expressed in radians. If \text{sin}(\text{\theta}) = \text{sin}(\text{\alpha}), then the general solution is \text{\theta} = n\pi + (-1)^n \text{\alpha}. If \text{cos}(\text{\theta}) = \text{cos}(\text{\alpha}), then \text{\theta} = 2n\pi \pm \text{\alpha}. If \text{tan}(\text{\theta}) = \text{tan}(\text{\alpha}), then \text{\theta} = n\pi + \text{\alpha}. These integers $n$ can be positive, negative, or zero.

Submultiple angles are used to find \text{sin}(\frac{\text{\theta}}{2}) and \text{cos}(\frac{\text{\theta}}{2}) when \text{cos}(\text{\theta}) or \text{sin}(\text{\theta}) is known. From absolute identities, \text{cos}(\frac{\text{\theta}}{2}) = \pm \text{\sqrt{\frac{1+\text{cos}(\text{\theta})}{2}}} and \text{sin}(\frac{\text{\theta}}{2}) = \pm \text{\sqrt{\frac{1-\text{cos}(\text{\theta})}{2}}}. The ambiguity in sign is resolved by the specific value or range of \text{\theta}. When given \text{sin}(\text{\theta}), one uses the relations \text{sin}(\frac{\text{\theta}}{2}) + \text{cos}(\frac{\text{\theta}}{2}) = \pm \text{\sqrt{1 + \text{sin}(\text{\theta})}} and \text{sin}(\frac{\text{\theta}}{2}) - \text{cos}(\frac{\text{\theta}}{2}) = \pm \text{\sqrt{1 - \text{sin}(\text{\theta})}} to determine the components. The formula for the third part of an angle, \text{cos}(\frac{\text{\theta}}{3}), involves roots of the cubic equation 4 \text{cos}^3(\frac{\text{\theta}}{3}) - 3 \text{cos}(\frac{\text{\theta}}{3}) = \text{cos}(\text{\theta}). General cubic equations $ax^3 + 3bx^2 + 3cx + d = 0$ can be solved using trigonometric substitutions when they possess three real roots.

Inverse trigonometric functions like $\text{sin}^{-1}(y)$, $\text{cos}^{-1}(k)$, and $\text{tan}^{-1}(k)$ represent many-valued functions. For a given value, the numerically least value is the principal value. Identities involving these functions include $\text{tan}^{-1}(m) \pm \text{tan}^{-1}(m') = \text{tan}^{-1}(\frac{m \pm m'}{1 \mp mm'})$. For principal values, the relation $\text{tan}^{-1}(m) + \text{tan}^{-1}(n) = k\pi + \text{tan}^{-1}(\frac{m+n}{1-mn})$ holds, where $k=0$ unless $mn > 1$, in which case $k$ is $1$ or $-1$ based on the quadrant rules.

Hyperbolic and Logarithmic Functions

The area-function for the rectangular hyperbola $y = \frac{1}{x}$ for $x > 0$ is defined as $hyp(t) = \int_1^t \frac{1}{x} \,dx$. This function is positive if $t > 1$ and negative if $0 < t < 1$. As $t \to \infty$, $hyp(t) \to \infty$, and as $t \to 0$, $hyp(t) \to - \infty$. The derivative is $\frac{d}{dt} hyp(t) = \frac{1}{t}$. Fundamental properties include $hyp(ab) = hyp(a) + hyp(b)$ and $hyp(a^n) = n \times hyp(a)$. This function provides the natural logarithm, $\text{log}_e(x)$. The value $e$ is defined such that $hyp(e) = 1$, which is approximately $2.71828$.

The exponential function $exp(y)$ is the inverse of the natural logarithm, such that if $y = \text{log}(x)$, then $x = exp(y) = e^y$. It satisfies the functional law $exp(y_1 + y_2) = exp(y_1) \times exp(y_2)$. Its derivative is $\frac{d}{dx} e^x = e^x$. Standard logarithmic inequalities include $1 - \frac{1}{t} < \text{log}(t) < t - 1$ for all positive $t \neq 1$, and $\frac{t}{1+t} < \text{log}(1+t) < t$ for $1+t > 0, t \neq 0$.

Limit theorems for these functions include $\lim_{x \to \infty} \frac{\text{log}(x)}{x^p} = 0$ for $p > 0$, and $\lim_{x \to 0} x \text{log}(x) = 0$. The value of $e$ can be expressed as $\lim_{n \to \infty} (1 + \frac{x}{n})^n = e^x$. Euler's Constant $\gamma$ is the limit as $n \to \infty$ of the sequence $u_n = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n} - \text{log}(n)$, which is approximately $0.577$.

Expansions in Power Series

Power series expressions represent functions as infinite sums. The series $1 - x + x^2 - x^3 + \dots$ converges to $\frac{1}{x+1}$ for $|x| < 1$. Fundamental expansions include the trigonometric series $\text{sin}(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots$ and $\text{cos}(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots$, both convergent for all real $x$ in radians. From these, expansions such as $\text{tan}(x) = x + \frac{x^3}{3} + \frac{2x^5}{15} + \dots$ are derived.

The logarithmic series $\text{log}(1+y) = y - \frac{y^2}{2} + \frac{y^3}{3} - \dots$ is convergent for $-1 < y \leq 1$. For numerical computation, the series $\frac{1}{2} \text{log}(y) = (\frac{y-1}{y+1}) + \frac{1}{3} (\frac{y-1}{y+1})^3 + \dots$ is used for $y > 0$. Gregory's Series for $\text{tan}^{-1}(x)$ is $\text{tan}^{-1}(y) = y - \frac{y^3}{3} + \frac{y^5}{5} - \dots$, which holds for $-1 \leq y \leq 1$ provided the principal value is taken within the range. Standard formulae for $\pi$ evaluation include Machin's form: $\frac{\pi}{4} = 4 \text{tan}^{-1}(\frac{1}{5}) - \text{tan}^{-1}(\frac{1}{239})$.

The exponential series $e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$ converges for all real and complex $x$. The value of $e$ is calculated from the sum of the reciprocals of factorials. The remainder after $n$ terms is bounded as $e - s_n < \frac{1}{(n-1)(n-1)!}$. The limit definition $\lim_{k \to \infty} (1 + \frac{1}{k})^k = e$ is equivalent to the series form.

Special Hyperbolic Functions

The hyperbolic sine and cosine are defined based on the exponential function as $sh(x) = \frac{1}{2}(e^x - e^{-x})$ and $ch(x) = \frac{1}{2}(e^x + e^{-x})$, also written as $\text{sinh}(x)$ and $\text{cosh}(x)$. They have the power series expansions $sh(x) = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \dots$ and $ch(x) = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \dots$. Other functions like $th(x) = sh(x) / ch(x)$ and $sech(x) = 1 / ch(x)$ are defined analogously to circular functions.

Fundamental identities for hyperbolic functions reflect those of circular functions but with specific sign changes according to Osborn's Rule. For example, $ch^2(x) - sh^2(x) = 1$ and $ch(x_1 + x_2) = ch(x_1) ch(x_2) + sh(x_1) sh(x_2)$. Differentiation results in $\frac{d}{dx} sh(x) = ch(x)$ and $\frac{d}{dx} ch(x) = sh(x)$, while $\frac{d}{dx} th(x) = \text{sech}^2(x)$. Inverse functions include sh^{-1}(y) = \text{log}[y + \text{\sqrt{y^2+1}}], ch^{-1}(y) = \pm \text{log}[y + \text{\sqrt{y^2-1}}] for $y \geq 1$, and $th^{-1}(y) = \frac{1}{2} \text{log}(\frac{1+y}{1-y})$ for $|y| < 1$.

Geometrically, coordinates of any point on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ can be written as (a \,ch(\text{\theta}), b \,sh(\text{\theta})). The area of the sector bounded by the curve, the x-axis, and the line from the origin to the point is \frac{1}{2} ab\text{\theta}. Hyperbolic functions are similarly useful in integration involving expressions like \text{\sqrt{a^2+x^2}} or \text{\sqrt{x^2-a^2}}.

Projection and Finite Series

The projection of a displacement on a line is the difference between the coordinates of the endpoints. If a line of length $l$ makes an angle \text{\theta} with an axis $Ox$, its projection is l \,cos(\text{\theta}). The projection of AC is the sum of the projections of $AB_1, B_1B_2, \dots, B_nC$. This principle allows for the derivation of addition theorems: $\text{cos}(A+B) = \text{cos}(A)\text{cos}(B) - \text{sin}(A)\text{sin}(B)$ and $\text{sin}(A+B) = \text{sin}(A)\text{cos}(B) + \text{cos}(A)\text{sin}(B)$.

Finite series of sines or cosines with angles in arithmetic progression can be summed via projection geometry or the difference method. The sum of the series \text{cos}(\text{\alpha}) + \text{cos}(\text{\alpha}+\text{\beta}) + \dots + \text{cos}(\text{\alpha} + (n-1)\text{\beta}) is given by \frac{\text{cos}(\text{\alpha} + \frac{n-1}{2}\text{\beta}) \times \text{sin}(\frac{n\text{\beta}}{2})}{\text{sin}(\frac{\text{\beta}}{2})}. Similarly, the sum of the sine series is \frac{\text{sin}(\text{\alpha} + \frac{n-1}{2}\text{\beta}) \times \text{sin}(\frac{n\text{\beta}}{2})}{\text{sin}(\frac{\text{\beta}}{2})}. These are remembered as $\text{cos}(\text{average angle}) \times \frac{\text{sin}(\text{n times semi-difference})}{\text{sin}(\text{semi-difference})}$.

The difference method involves expressing the general term $u_r$ as $f(r+1) - f(r)$, leading to the telescopic sum $f(n+1) - f(1)$. For example, the series \sum 2^{r-1} \text{tan}(2^{r-1} \text{\theta}) can be summed since \text{tan}(\text{\theta}) = \text{cot}(\text{\theta}) - 2 \text{cot}(2\text{\theta}). This method often requires identifying functional forms where differences simplify the summation process.

Complex Numbers and the Argand Diagram

A complex number is defined as an ordered pair of real numbers $[a, b]$, denoted as $a + ib$, where $i = [0, 1]$ and $i^2 = -1$. Addition follows the rule $[a, b] + [c, d] = [a+c, b+d]$, while multiplication is defined as $[a, b] \times [c, d] = [ac - bd, ad + bc]$. Division by a non-zero complex number $c + id$ gives $[a, b] \div [c, d] = [\frac{ac + bd}{c^2 + d^2}, \frac{bc - ad}{c^2 + d^2}]$.

The Argand Diagram represents complex numbers as points in a plane. The modulus of $z = x + iy$ is |z| = r = \text{\sqrt{x^2 + y^2}}, and the amplitude is \text{\theta} such that x = r \,cos(\text{\theta}) and y = r \,sin(\text{\theta}). The standard form is z = r(\text{cos}(\text{\theta}) + i \text{sin}(\text{\theta})) or r \,cis(\text{\theta}). For any complex number, the relative position is defined by its modulus and unique principal amplitude \text{\theta} \in (-\pi, \pi]. Triangle inequalities state that $|z_1 + z_2| \leq |z_1| + |z_2|$ and $|z_1 + z_2| \geq ||z_1| - |z_2||$.

Product and quotient rules in polar form reveal that the modulus of a product is the product of the moduli, and its amplitude is the sum of the amplitudes. That is, $|z_1 z_2| = |z_1| \times |z_2|$ and $am(z_1 z_2) = am(z_1) + am(z_2)$. Logarithmically, this means $\text{am}(z_1 \cdot z_2) = \text{am}(z_1) + \text{am}(z_2)$. Inversions and other transformations map points on circles or lines to other points, illustrating geometric properties such as the cardioid locus derived from $OP = (OA \cdot OQ)^{1/2}$.

De Moivre's Theorem and Applications

De Moivre's Theorem states that for any rational number $n$, \text{cos}(n\text{\theta}) + i \text{sin}(n\theta) is a value of (\text{cos}(\text{\theta}) + i \text{sin}(\text{\theta}))^n. For integral $n$, this is the unique value. For fractional $n = p/q$, there are $q$ distinct values given by cis(\frac{p\text{\theta} + 2k\pi}{q}) for $k = 0, 1, \dots, q-1$. These values represent vertices of a regular polygon inscribed in the unit circle. Powers of complex numbers are defined using the relation $z^w = exp(w \,Log \,z)$.

Expansions of powers of circular functions into multiple angles utilize the relations 2 \,cos(\text{\theta}) = z + z^{-1} and 2i \,sin(\text{\theta}) = z - z^{-1}. By the binomial theorem, (2 \,cos \text{\theta})^n = z^n + n z^{n-2} + \dots + z^{-n}, which allows simplification to sums of terms like 2 \,cos(k\text{\theta}). Conversely, \text{cos}(n\text{\theta}) and \text{sin}(n\text{\theta}) can be expanded as polynomials in terms of \text{cos}(\text{\theta}) and \text{sin}(\text{\theta}). For integer $n$, \text{cos}(n\text{\theta}) = \text{cos}^n \text{\theta} - \binom{n}{2} \text{cos}^{n-2} \text{\theta} \,sin^2 \text{\theta} + \dots.

Summation of series like 1 + x \text{cos}(\text{\theta}) + x^2 \text{cos}(2\text{\theta}) + \dots is achieved by considering the real part of the complex geometric progression. The sum to infinity for $|x| < 1$ is \frac{1 - x \text{cos}(\text{\theta})}{1 - 2x \text{cos}(\text{\theta}) + x^2}. For sines, the sum is \frac{x \,sin(\text{\theta})}{1 - 2x \,cos(\text{\theta}) + x^2}. These results extend to complex variables within their circles of convergence.

Functions of a Complex Variable

The exponential function of a complex variable is defined as $exp(z) = \lim_{n \to \infty} (1 + \frac{z}{n})^n = \sum \frac{z^n}{n!}$. For $z = x + iy$, $exp(z) = e^x(\text{cos}(y) + i \text{sin}(y))$. This indicates that $exp(z)$ is a periodic function with a period of $2\pi i$. Generalised definitions for circular and hyperbolic functions follow: $cos(z) = \frac{1}{2}(e^{iz} + e^{-iz})$ and $sin(z) = \frac{1}{2i}(e^{iz} - e^{-iz})$, which maintain standard trigonometric identities.

Logarithms of complex numbers are defined such that if $w = exp(z)$, then $z = Log(w)$. This is an infinitely many-valued function given by Log(\text{\rho} \cdot cis(\text{\phi})) = \text{log}(\text{\rho}) + i(\text{\phi} + 2n\pi). The principal value $\text{log}(w)$ uses the principal amplitude in $(-\pi, \pi]$. Inverse functions for complex variables, such as $Tan^{-1}(x+iy)$, are calculated by separating real and imaginary parts. Specifically, Tan^{-1}(x+iy) = n\pi + \text{\alpha} + \frac{i}{4} \text{log} \frac{x^2 + (1+y)^2}{x^2 + (1-y)^2}.

Roots and Factors

Formation of equations with assigned roots, such as $\text{cos}(\frac{2r\pi}{n})$, allows the derivation of many identities. For example, the roots of $8c^3 + 4c^2 - 4c - 1 = 0$ correspond to $\text{cos}(\frac{2\pi}{7})$, $\text{cos}(\frac{4\pi}{7})$, and $\text{cos}(\frac{6\pi}{7})$. Symmetric functions of roots provide values for sums like $\sum \sec(\frac{2r\pi}{7})$. The sum $\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$ is derived from properties of \text{sin} \,n\text{\theta}.

Polynomials like $x^n - 1$ and $x^n + 1$ have factorisations into real quadratic factors. $x^n - 1$ for even $n$ yields $(x-1)(x+1) \prod_{k=1}^{n/2-1} (x^2 - 2x \text{cos}(\frac{2k\pi}{n}) + 1)$. Trigonometric functions such as \text{sin}(n\text{\theta}) also exhibit factorisations founded on their zeroes: \text{sin}(n\text{\theta}) = n \text{sin}(\text{\theta}) \prod_{r=1}^{(n-1)/2} (1 - \frac{\text{sin}^2(\text{\theta})}{\text{sin}^2(\frac{r\pi}{n})}) when $n$ is odd.