Geometry and Trigonometry

40 Cartesian & Polar Co-ordinates

In mathematics, coordinate systems are used to specify the position of points in space. Two of the most common systems are Cartesian coordinates and Polar coordinates, each offering unique advantages depending on the problem.

40.1 Introduction

The Cartesian coordinate system (also known as the rectangular coordinate system) uses two perpendicular axes (x and y) to define a point's position as an ordered pair $(x, y)$. This is the most familiar system for graphing linear and quadratic equations.

The Polar coordinate system defines a point's position by its distance from a fixed point (the pole or origin) and the angle it makes with a fixed direction (the polar axis). A point is specified by $(r, \theta)$, where $r$ is the radial distance and $\theta$ is the angular position.

40.2 Cartesian to Polar Conversion

To convert a point from Cartesian coordinates $(x, y)$ to Polar coordinates $(r, \theta)$:

• Radius ($r$): Use the Pythagorean theorem: $$ r = \sqrt{x^2 + y^2} $$
• Angle ($\theta$): Use the inverse tangent function, being mindful of the quadrant to get the correct angle: $$ \theta = \arctan\left(\frac{y}{x}\right) $$ You might need to add $180^\circ$ (or $\pi$ radians) if $x$ is negative, or $360^\circ$ (or $2\pi$ radians) if $\theta$ ends up negative and you want it in $[0, 360^\circ)$ range. It's often safer to use `Math.atan2(y, x)` function available in many programming languages and calculators, which automatically handles quadrants.

The angle $\theta$ is typically given in radians or degrees, often in the range $0 \le \theta < 2\pi$ or $0^\circ \le \theta < 360^\circ$.

40.3 Polar to Cartesian Conversion

To convert a point from Polar coordinates $(r, \theta)$ to Cartesian coordinates $(x, y)$:

• X-coordinate ($x$): $$ x = r \cos \theta $$
• Y-coordinate ($y$): $$ y = r \sin \theta $$

40.4 Calculator Pol/Rec Functions

Most scientific calculators have built-in functions for converting between Cartesian and Polar coordinates, often labeled as "Pol(" (for Cartesian to Polar) and "Rec(" (for Polar to Cartesian). These functions handle the quadrant adjustments automatically, simplifying calculations.

Example Calculator Usage:

• To convert Cartesian $(3, 4)$ to Polar: Type `Pol(3, 4)` and press ENTER. It might return `r=5`, `theta=53.13`.
• To convert Polar $(5, 30^\circ)$ to Cartesian: Type `Rec(5, 30)` and press ENTER. It might return `x=4.33`, `y=2.5`.

Always ensure your calculator is in the correct angle mode (DEG or RAD) before using these functions.

41 Non-Right-Angled Triangles

While basic trigonometry focuses on right-angled triangles, many real-world problems involve triangles without a $90^\circ$ angle. For these, we use the Sine Rule and Cosine Rule.

41.1 Sine & Cosine Rules

Sine Rule:

The Sine Rule relates the sides of a triangle to the sines of their opposite angles. It is used when you know:

• Two angles and one side (AAS or ASA).
• Two sides and a non-included angle (SSA - ambiguous case).

For any triangle ABC with sides $a, b, c$ opposite to angles $A, B, C$ respectively:

Cosine Rule:

The Cosine Rule is a generalization of the Pythagorean theorem. It is used when you know:

• Two sides and the included angle (SAS).
• All three sides (SSS).

For any triangle ABC:

It can also be rearranged to find an angle when all three sides are known:

41.2 Area of Any Triangle

The area of any triangle can be calculated if you know the lengths of two sides and the measure of the angle included between them:

Or similarly:

41.3 Triangle Solution Problems

Solving non-right-angled triangles involves determining all unknown angles and side lengths given a set of initial information. The choice between the Sine Rule and Cosine Rule depends on the given information:

• Use Sine Rule if: You have an angle-side pair (a known angle and its opposite side), and one other piece of information (another angle or another side). This covers AAS, ASA, and SSA (ambiguous case).
• Use Cosine Rule if: You have three sides (SSS) and want to find an angle, or two sides and the included angle (SAS) and want to find the third side.

42 Trigonometric Identities & Equations

Trigonometric identities are equations that are true for all values of the variables for which the expressions are defined. Trigonometric equations are equations that involve trigonometric functions and are only true for specific values of the variable.

42.1 Trigonometric Identities

Identities are powerful tools for simplifying expressions and solving equations. Here are some of the most common categories:

Fundamental Identities:

• Reciprocal Identities: $$ \csc \theta = \frac{1}{\sin \theta} $$ $$ \sec \theta = \frac{1}{\cos \theta} $$ $$ \cot \theta = \frac{1}{\tan \theta} $$
• Quotient Identities: $$ \tan \theta = \frac{\sin \theta}{\cos \theta} $$ $$ \cot \theta = \frac{\cos \theta}{\sin \theta} $$
• Pythagorean Identities: $$ \sin^2 \theta + \cos^2 \theta = 1 $$ $$ 1 + \tan^2 \theta = \sec^2 \theta $$ $$ 1 + \cot^2 \theta = \csc^2 \theta $$

Compound Angle Formulae (Addition/Subtraction Formulae):

• $$ \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B $$
• $$ \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B $$
• $$ \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} $$

Double Angle Formulae:

• $$ \sin 2A = 2 \sin A \cos A $$
• $$ \cos 2A = \cos^2 A - \sin^2 A = 2 \cos^2 A - 1 = 1 - 2 \sin^2 A $$
• $$ \tan 2A = \frac{2 \tan A}{1 - \tan^2 A} $$

42.2 Identity Problems

Proving trigonometric identities involves transforming one side of the equation (usually the more complex side) into the other side using known identities and algebraic manipulation. Key strategies include:

• Start with the more complicated side.
• Express everything in terms of sine and cosine.
• Look for opportunities to use Pythagorean identities.
• Combine fractions or split single fractions.
• Factor expressions or expand terms.
• Multiply by a conjugate.

42.3 Trigonometric Equations

Trigonometric equations are equations that contain trigonometric functions of unknown angles. Unlike identities, these equations are only true for specific values of the angle. When solving, remember to consider:

• The basic angle (the acute angle).
• The quadrant(s) where the trigonometric function has the required sign (using the CAST diagram).
• The periodicity of the function to find all possible solutions (general solution).
• The specified range for the solutions (e.g., $0^\circ \le \theta < 360^\circ$).

The general solutions for basic equations are:

• For $\sin \theta = k$: $$ \theta = n \cdot 360^\circ + \alpha \quad \text{or} \quad \theta = n \cdot 360^\circ + (180^\circ - \alpha) $$ where $\alpha = \sin^{-1}(k)$ and $n$ is an integer.
• For $\cos \theta = k$: $$ \theta = n \cdot 360^\circ \pm \alpha $$ where $\alpha = \cos^{-1}(k)$ and $n$ is an integer.
• For $\tan \theta = k$: $$ \theta = n \cdot 180^\circ + \alpha $$ where $\alpha = \tan^{-1}(k)$ and $n$ is an integer.

42.4 Equation Problems (i)

These are the simplest forms, where you directly find the basic angle and then use the CAST diagram to find solutions in the given range. For example, solving $\sin x = 0.5$ or $\cos x = -0.8$.

42.5 Equation Problems (ii)

These involve transformations of the angle, like $\sin(2x) = k$ or $\cos(x - 30^\circ) = k$. The key is to transform the range accordingly before finding solutions for the composite angle.

42.6 Equation Problems (iii)

This category includes equations that are quadratic in form (e.g., $2\sin^2 x + \sin x - 1 = 0$) or require the use of identities to simplify them into a solvable form (e.g., using $\sin^2 x + \cos^2 x = 1$ to eliminate one trigonometric function).

42.7 Equation Problems (iv)

These are more complex equations that might require multiple steps of identity application, factoring, or rearrangement to solve. They often combine concepts from previous sections, such as using double angle identities to reduce an equation to a simpler form.

43 Trigonometric & Hyperbolic Relations

Trigonometric and hyperbolic functions, while seemingly distinct, share deep mathematical connections, particularly when explored in the realm of complex numbers. They both arise from the geometry of a unit circle and a unit hyperbola, respectively, and are closely related through Euler's formula.

43.1 The Relationship (Using Euler's Formula)

The relationship between trigonometric functions (like sine and cosine) and the exponential function (which forms the basis for hyperbolic functions) is beautifully expressed by Euler's Formula:

Where $i$ is the imaginary unit ($i^2 = -1$).

From Euler's formula, we can derive expressions for sine and cosine in terms of complex exponentials:

• By replacing $x$ with $-x$ in Euler's formula: $$ e^{-ix} = \cos(-x) + i \sin(-x) = \cos x - i \sin x $$
• Adding $e^{ix}$ and $e^{-ix}$: $$ e^{ix} + e^{-ix} = (\cos x + i \sin x) + (\cos x - i \sin x) = 2 \cos x $$ Therefore, $$ \cos x = \frac{e^{ix} + e^{-ix}}{2} $$
• Subtracting $e^{-ix}$ from $e^{ix}$: $$ e^{ix} - e^{-ix} = (\cos x + i \sin x) - (\cos x - i \sin x) = 2i \sin x $$ Therefore, $$ \sin x = \frac{e^{ix} - e^{-ix}}{2i} $$

Now, let's look at the definitions of the hyperbolic functions:

• Hyperbolic Cosine (cosh): $$ \cosh x = \frac{e^x + e^{-x}}{2} $$
• Hyperbolic Sine (sinh): $$ \sinh x = \frac{e^x - e^{-x}}{2} $$

By comparing these, we can see the direct relationships:

• $$ \cos(ix) = \frac{e^{i(ix)} + e^{-i(ix)}}{2} = \frac{e^{-x} + e^{x}}{2} = \cosh x $$
• $$ \sin(ix) = \frac{e^{i(ix)} - e^{-i(ix)}}{2i} = \frac{e^{-x} - e^{x}}{2i} = \frac{-(e^x - e^{-x})}{2i} = \frac{-2i \sinh x}{2i} = i \sinh x $$

These fundamental relationships ($ \cos(ix) = \cosh x $ and $ \sin(ix) = i \sinh x $) reveal the deep connection between the two families of functions.

43.2 Hyperbolic Identities

Similar to trigonometric identities, hyperbolic functions also have a set of identities that can be derived from their definitions. Many of these mirror trigonometric identities, with some sign changes.

Fundamental Hyperbolic Identities:

• $$ \cosh^2 x - \sinh^2 x = 1 $$ (This is the hyperbolic equivalent of $\sin^2 x + \cos^2 x = 1$)
• $$ 1 - \tanh^2 x = \text{sech}^2 x $$
• $$ \coth^2 x - 1 = \text{csch}^2 x $$

Other common hyperbolic functions:

• Hyperbolic Tangent (tanh): $$ \tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}} $$
• Hyperbolic Secant (sech): $$ \text{sech } x = \frac{1}{\cosh x} = \frac{2}{e^x + e^{-x}} $$
• Hyperbolic Cosecant (csch): $$ \text{csch } x = \frac{1}{\sinh x} = \frac{2}{e^x - e^{-x}} $$
• Hyperbolic Cotangent (coth): $$ \coth x = \frac{1}{\tanh x} = \frac{e^x + e^{-x}}{e^x - e^{-x}} $$

44 Compound Angles

Compound angle formulas allow us to express trigonometric functions of sums or differences of angles in terms of trigonometric functions of the individual angles. These are powerful tools for simplifying expressions, solving equations, and analyzing waveforms.

44.1 Compound Angle Formulae

These identities are fundamental for working with compound angles:

• $$ \sin(A+B) = \sin A \cos B + \cos A \sin B $$
• $$ \sin(A-B) = \sin A \cos B - \cos A \sin B $$
• $$ \cos(A+B) = \cos A \cos B - \sin A \sin B $$
• $$ \cos(A-B) = \cos A \cos B + \sin A \sin B $$
• $$ \tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} $$
• $$ \tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} $$

44.2 Converting $a \sin \omega t + b \cos \omega t$

Expressions of the form $a \sin \theta + b \cos \theta$ can be converted into a single trigonometric function of the form $R \sin(\theta \pm \alpha)$ or $R \cos(\theta \pm \alpha)$, where $R$ is the amplitude and $\alpha$ is the phase angle. This is particularly useful in physics and engineering for analyzing oscillating systems.

To convert $a \sin \theta + b \cos \theta$ to $R \sin(\theta + \alpha)$:

• Find $R$: $$ R = \sqrt{a^2 + b^2} $$
• Find $\alpha$: Compare $a \sin \theta + b \cos \theta$ with $R \sin \theta \cos \alpha + R \cos \theta \sin \alpha$. This gives $a = R \cos \alpha$ and $b = R \sin \alpha$. Therefore, $$ \tan \alpha = \frac{R \sin \alpha}{R \cos \alpha} = \frac{b}{a} $$ Be careful to find $\alpha$ in the correct quadrant based on the signs of $a$ and $b$.

Similarly, for $R \cos(\theta - \alpha)$, you would compare $a \sin \theta + b \cos \theta$ with $R \cos \theta \cos \alpha + R \sin \theta \sin \alpha$, leading to $\tan \alpha = \frac{a}{b}$.

44.3 Double Angles

Double angle identities are special cases of compound angle formulae where $A = B$.

• $$ \sin 2A = 2 \sin A \cos A $$
• $$ \cos 2A = \cos^2 A - \sin^2 A $$ Also: $$ \cos 2A = 2 \cos^2 A - 1 $$ And: $$ \cos 2A = 1 - 2 \sin^2 A $$
• $$ \tan 2A = \frac{2 \tan A}{1 - \tan^2 A} $$

44.4 Products to Sums/Differences

These identities are used to transform products of sines and cosines into sums or differences, which can simplify integration or other calculations.

• $$ 2 \sin A \cos B = \sin(A+B) + \sin(A-B) $$
• $$ 2 \cos A \sin B = \sin(A+B) - \sin(A-B) $$
• $$ 2 \cos A \cos B = \cos(A+B) + \cos(A-B) $$
• $$ 2 \sin A \sin B = \cos(A-B) - \cos(A+B) $$

44.5 Sums/Differences to Products

These identities are the inverse of the product-to-sum identities and are useful for factoring trigonometric expressions or solving certain types of equations.

• $$ \sin P + \sin Q = 2 \sin\left(\frac{P+Q}{2}\right) \cos\left(\frac{P-Q}{2}\right) $$
• $$ \sin P - \sin Q = 2 \cos\left(\frac{P+Q}{2}\right) \sin\left(\frac{P-Q}{2}\right) $$
• $$ \cos P + \cos Q = 2 \cos\left(\frac{P+Q}{2}\right) \cos\left(\frac{P-Q}{2}\right) $$
• $$ \cos P - \cos Q = -2 \sin\left(\frac{P+Q}{2}\right) \sin\left(\frac{P-Q}{2}\right) $$

44.6 Power Waveforms (AC Circuits)

In AC (Alternating Current) circuits, voltage and current are often represented by sinusoidal waveforms. The power delivered in an AC circuit is the product of voltage and current. When voltage and current are out of phase, their product leads to a power waveform that can be simplified using product-to-sum identities. This helps in understanding average power, reactive power, and apparent power, which are critical concepts in electrical engineering.