Topic 8: Trigonometry
Welcome to this introductory segment focusing on Trigonometry. Derived from Greek words that translate to 'triangle measurement,' this branch of mathematics is fundamentally concerned with exploring and quantifying the relationships between the angles and side lengths of triangles. Initially, its primary focus lay on right-angled triangles, establishing principles that govern their unique properties. However, the evolution of the field has significantly expanded its purview, extending far beyond simple triangular analysis to become an indispensable tool for understanding and modeling diverse phenomena, particularly those exhibiting periodic behaviour, which are prevalent throughout science and engineering disciplines.
The cornerstone of elementary trigonometry is built upon the concept of trigonometric ratios. For an acute angle within a right-angled triangle, six fundamental ratios are defined based on the specific proportions of the lengths of its sides: the side opposite the angle, the side adjacent to the angle, and the hypotenuse (the side opposite the right angle). These ratios are known as sine (sin), cosine (cos), tangent (tan), and their respective reciprocals: cosecant (csc), secant (sec), and cotangent (cot). A crucial step in mastering these basics involves becoming familiar with the values of these ratios for standard, frequently encountered angles such as 0°, 30°, 45°, 60°, and 90°, as these values are often used directly or as building blocks in calculations.
A significant component of trigonometric study involves understanding and skillfully manipulating trigonometric identities. These are equations involving trigonometric ratios that hold true for all valid input values of the variables (angles). Fundamental identities form the basis for simplifying expressions and solving equations. Key examples include the reciprocal identities (e.g., $\sin \theta = \frac{1}{\csc \theta}$), quotient identities (e.g., $\tan \theta = \frac{\sin \theta}{\cos \theta}$), and the foundational Pythagorean identity ($\sin^2 \theta + \cos^2 \theta = 1$). These identities are vital tools for transforming trigonometric expressions and solving related problems.
The concept of angles is generalized beyond the acute range through the introduction of the unit circle. This geometric construct allows trigonometric ratios, which are then often referred to as trigonometric functions, to be defined for any angle. This includes negative angles and angles exceeding 360° (or $2\pi$ radians when using radian measure), accommodating a full range of rotational possibilities. This generalization leads to the detailed study of trigonometric functions and their characteristic graphs. These graphs reveal the inherent periodic or wave-like nature of these functions, a property that makes trigonometry exceptionally powerful for modeling oscillating systems, wave phenomena (like light and sound), and other cyclical processes observed in the natural and engineered world. The concept of inverse trigonometric functions is also introduced, allowing us to work backward from a given ratio value to find the corresponding angle.
While initially focused on right triangles, trigonometry also provides essential tools for solving triangles that do not contain a right angle – known as oblique triangles. The primary instruments for this are the Sine Rule and the Cosine Rule. These rules establish relationships between the sides and angles of any triangle, enabling us to calculate unknown sides or angles given sufficient information. One of the most direct and easily graspable practical applications of trigonometry is in solving problems related to heights and distances. By measuring angles of elevation or depression, trigonometry allows us to calculate lengths or heights that are otherwise inaccessible or difficult to measure directly, such as the height of a tall building or the distance across a river.
The utility of trigonometry extends across an astonishingly broad spectrum of fields. It is absolutely indispensable in physics (especially in mechanics, waves, optics, acoustics), various branches of engineering (electrical, mechanical, civil), navigation (from ancient sailing to modern GPS), surveying (for mapping and land measurement), astronomy (for calculating positions and distances of celestial bodies), music theory (relating to sound waves), and even computer graphics (for transformations and projections). Its ability to relate angles, lengths, and periodic behavior makes it a fundamental mathematical tool in quantitative analysis and problem-solving in countless scientific and technical domains.
Introduction to Trigonometry and Basic Ratios (Right Triangle Trigonometry)
Trigonometry is the study of relationships between the sides and angles of triangles, initially focusing on Right Triangles. Key terms relate angles to sides: opposite, adjacent, and hypotenuse. The six fundamental Trigonometrical Ratios (Sine, Cosine, Tangent, Cosecant, Secant, Cotangent) are defined as ratios of these side lengths (e.g., $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$, $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$, $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$). We explore essential Relationships between Ratios, including reciprocal identities ($\csc \theta = \frac{1}{\sin \theta}$) and quotient identities ($\tan \theta = \frac{\sin \theta}{\cos \theta}$).
Trigonometric Ratios of Special Angles and Complementary Angles
We calculate and memorise the exact values of trigonometric ratios for certain Specific Angles like $0^\circ, 30^\circ, 45^\circ, 60^\circ,$ and $90^\circ$. For instance, $\sin 45^\circ = \frac{1}{\sqrt{2}}$ and $\tan 60^\circ = \sqrt{3}$. Understanding the relationship between Complementary Angles ($A$ and $90^\circ - A$) is also crucial. The trigonometric ratios of complementary angles are related; for example, $\sin(90^\circ - A) = \cos A$, $\tan(90^\circ - A) = \cot A$, and $\sec(90^\circ - A) = \csc A$. These values and relationships are fundamental for solving trigonometric problems.
Fundamental Trigonometric Identities
Trigonometric Identities are equations involving trigonometric ratios that are true for all valid values of the angle. These identities are powerful tools for simplification. We cover the fundamental Reciprocal Identities (e.g., $\sec \theta = \frac{1}{\cos \theta}$) and the Quotient Identities (e.g., $\cot \theta = \frac{\cos \theta}{\sin \theta}$). The most important are the three Pythagorean Identities: $\sin^2 \theta + \cos^2 \theta = 1$, $1 + \tan^2 \theta = \sec^2 \theta$, and $1 + \cot^2 \theta = \csc^2 \theta$. These identities are used extensively for Simplifying Expressions and Proving Other Identities.
Angle Measurement: Degrees and Radians
Angles can be measured using different systems. The familiar Degree Measure divides a circle into $360^\circ$. The Radian Measure is an alternative unit where an angle is defined by the ratio of the arc length it subtends to the radius. One radian is the angle subtended by an arc equal in length to the radius. The relationship is $\pi \text{ radians} = 180^\circ$. We learn the process of Conversion between Degree and Radian Measure ($\text{radians} = \text{degrees} \times \frac{\pi}{180^\circ}$). Radian measure simplifies formulas for the Length of an Arc ($L=r\theta$) and the Area of a Sector ($A=\frac{1}{2}r^2\theta$, where $\theta$ is in radians).
Trigonometric Functions of a Real Number (Unit Circle Approach)
Using the Unit Circle (a circle of radius 1 centered at the origin), we extend the definition of trigonometric ratios to Trigonometric Functions of Any Real Number (angle). The coordinates $(x, y)$ where the angle's terminal side intersects the unit circle define the functions: $\cos \theta = x$, $\sin \theta = y$. This approach helps understand the Domain and Range of each function, the Signs of Functions in Different Quadrants (e.g., $\sin \theta > 0$ in Quadrants I and II), and the Periodicity, showing how values repeat every $2\pi$ (or $360^\circ$) for sine and cosine, and every $\pi$ ($180^\circ$) for tangent.
Graphs of Trigonometric Functions
Visualizing trigonometric functions through their graphs reveals their properties. The Graph of the Sine Function ($y=\sin x$) and Cosine Function ($y=\cos x$) are continuous, periodic waves with amplitude 1 and period $2\pi$. The Graph of the Tangent Function ($y=\tan x$) has a period of $\pi$ and vertical asymptotes. We also plot the Graphs of Reciprocal Functions ($y=\csc x, y=\sec x, y=\cot x$), which have their own characteristic shapes, periods, and asymptotes. Analyzing these graphs helps understand concepts like periodicity, amplitude, and phase shifts (horizontal translations).
Trigonometric Identities: Compound and Multiple Angles
Expanding on fundamental identities, we study those for Compound Angles (sums and differences of angles) and Multiple Angles. Formulas for $\sin(A \pm B)$, $\cos(A \pm B)$, and $\tan(A \pm B)$ are fundamental. These lead to Double Angle Formulas like $\cos 2A = 2\cos^2 A - 1$ and $\tan 2A = \frac{2\tan A}{1 - \tan^2 A}$. Further identities cover Half Angle and Triple Angle Formulas, often derived from compound and double angle formulas. These identities are vital for simplifying expressions and solving complex trigonometric equations.
Trigonometric Transformations: Product-to-Sum and Sum-to-Product
This section focuses on identities that convert products of trigonometric functions into sums or differences, and vice versa. Product-to-Sum Formulas, like $2\sin A \cos B = \sin(A+B) + \sin(A-B)$, are derived from compound angle formulas. Conversely, Sum-to-Product Formulas, such as $\sin A + \sin B = 2\sin(\frac{A+B}{2})\cos(\frac{A-B}{2})$, allow converting sums/differences into products. These transformations are powerful tools with significant Applications in simplifying complex expressions, proving other trigonometric identities, and solving certain types of trigonometric equations.
Trigonometric Equations: Solving Methods
A Trigonometric Equation is an equation involving trigonometric functions of a variable angle. Solving involves finding all possible angle values that satisfy the equation. We distinguish between Principal Solutions (typically within $[0, 2\pi)$ or $[0^\circ, 360^\circ)$) and General Solutions, which account for the functions' periodicity using integer $n$. We learn formulas for the General Solutions of basic equations like $\sin x = k$, $\cos x = k$, and $\tan x = k$. More complex equations are solved by Factoring, using Identities to simplify, or transforming them into basic solvable forms.
Inverse Trigonometric Functions: Introduction and Properties
Inverse Trigonometric Functions (denoted as $\sin^{-1}x, \cos^{-1}x$, etc.) are the inverses of the trigonometric functions, outputting the angle whose sine, cosine, etc., is $x$. Due to the periodic nature of trig functions, their inverses are restricted to specific ranges called Principal Value Branches to be single-valued. Understanding the specific Domain and Range of each inverse function's principal branch is crucial. We study various Properties, including sum/difference formulas and Conversion properties, which are essential for Simplification of Expressions involving inverse trigonometric functions and solving related equations.
Applications of Trigonometry: Heights and Distances
One of the most common practical Applications of Trigonometry is in solving problems involving Heights and Distances. This requires understanding Basics like the Angle of Elevation (looking upwards) and the Angle of Depression (looking downwards) relative to the horizontal Line of Sight. Problems are typically solved by forming one or more right-angled triangles from the given scenario. We apply trigonometric ratios (sine, cosine, tangent) to find unknown lengths or angles, tackling problems involving a Single Right Triangle initially, then progressing to scenarios requiring the analysis of Multiple Right Triangles.