Trigonometry: Right Triangles & Graphs

The foundations of trigonometry are traced back to ancient civilizations in Mesopotamia, Greece, and India. These cultures utilized the relationships between angles and lengths to better understand the movements of celestial bodies in the night sky. Early trigonometry heavily relied on angles and triangles inscribed within circles, with calculations often based on the lengths of chords. The core concept, the relationship between angles and distances, formed the basis for future developments in the field. The text highlights the dependence on geometric relationships within circles as a crucial component of early trigonometric calculations.

The development of key trigonometric functions is described, with Indian mathematicians calculating the earliest known values for the sine function around the 5th century. Subsequently, Islamic mathematicians expanded the field in the 11th century, adding cosine, tangent, cotangent, secant, and cosecant to the existing trigonometric toolkit. These advancements proved invaluable to European navigators during the Middle Ages, aiding in distance and direction calculations. The text explicitly points to the significant contributions of different cultures over time, emphasizing the evolution of the field from its early geometric focus to the inclusion of multiple trigonometric functions.

The use of degrees to measure angles is attributed to the ancient Mesopotamians, who assigned 360 degrees to a complete circle. The text suggests a possible connection between this assignment and the approximate number of days in a year (365), possibly linked to the observation of the Earth's journey around the Sun. The Mesopotamian system also incorporated subdivisions of degrees into minutes and seconds, with 60 seconds composing a minute and 60 minutes forming a degree. This section highlights a pivotal cultural influence in the standardization of angle measurement, a cornerstone of trigonometric calculations.

This section introduces the six fundamental trigonometric ratios: sine, cosine, tangent, cosecant, secant, and cotangent. These ratios define the relationships between the angles and sides of a right-angled triangle. The three primary ratios (sine, cosine, and tangent) are defined, along with their reciprocals. The text emphasizes the importance of these ratios as tools for solving triangles, which is the process of finding unknown angles and sides given some initial information about the triangle. Before the widespread use of calculators, these values were commonly found in tables within textbooks. Now, however, calculators, using methods studied in calculus, have largely replaced the need for these tables.

The process of 'solving triangles' is explained, highlighting its use in determining unknown sides and angles using known information. The Pythagorean theorem is presented as a key tool for finding the third side of a right triangle given the other two. The relationship between the two acute angles in a right triangle, noting that they are complements and sum to 90 degrees, is also mentioned. The text further explains that solving triangles is essential for indirect measurement—a technique used to find inaccessible distances. Examples given include measuring the height of a tall tree or the width of a body of water, demonstrating the practical application of trigonometry in real-world scenarios. The concept of the angle of elevation is introduced in this section; it's the angle between the horizontal and the line of sight to an object above the observer.

This section explores more complex applications of right triangle trigonometry, including problems that require the use of systems of equations. One example is determining the height of an object with an inaccessible base by measuring angles of elevation from multiple points. Other examples include using angles of depression and elevation to calculate distances, with scenarios involving guy wires attached to towers, a woman observing a building from a hill, and hot air balloons estimating their altitude. Specific numerical problems are provided, demonstrating the practical use of right triangle trigonometry and angles of elevation and depression. The use of trigonometric ratios to solve for unknown sides and angles in real-world scenarios forms the heart of this section.

This section extends the definitions of trigonometric functions (sine and cosine) to angles beyond the acute angles (0-90 degrees) found in right-angled triangles. It explains how to determine sine and cosine values for angles in all four quadrants of the coordinate plane. The concept of reference angles is introduced—the acute angle formed between the terminal side of an angle and the x-axis. The values of sine and cosine for quadrantal angles (0, 90, 180, 270, and 360 degrees) are explicitly stated. The method involves using a unit circle; a circle with a radius of 1, where the x and y coordinates of a point on the circle's circumference correspond to the cosine and sine of the angle formed by the radius connecting that point to the origin and the positive x-axis. By understanding the quadrant, the reference angle can be used to determine the sign of the trigonometric function and its value. The text illustrates this with examples in the second quadrant, highlighting how the supplementary angle within the created triangle can help find the side lengths and hence the x and y coordinates.

This section delves into graphing trigonometric functions, primarily focusing on the sine function. The graph's characteristics are described, including its starting point at zero, its rise to a maximum value of 1, its decrease back to zero, its descent into negative values to a minimum, and its eventual return to zero to complete one cycle. The concept of the period of the graph is defined—the length of one complete cycle. The standard period for sine and cosine functions is 2π radians. The impact of coefficients on the graph's transformations is discussed. In the general equation y = A sin(Bx), A affects the amplitude (the maximum distance from the center line), and B affects the period (making it 2π/B). The text also introduces the idea of determining an equation from a given graph by analyzing the amplitude and period, which are used to find A and B, to create the trigonometric equation. The section mentions that these graphs can be useful in modelling many wave-like phenomena.

The final subsection discusses phase shift, a horizontal displacement in the graph of a trigonometric function. This transformation is shown within the standard equation y = A sin(Bx + C) + D, where C represents the phase shift. It explains how the phase shift alters the starting point of the graph, influencing the x-values that correspond to the maximum, minimum, and zero points. The text emphasizes that to graph one period, at least five critical x-values (quadrantal angles) are needed. The effect of the phase shift on finding these critical points is clearly explained with an example involving the sine function, demonstrating how to find the new starting point and the intervals between critical points on the x-axis.

This section focuses on trigonometric identities—equations that are true for all values of the angles involved. The process of verifying these identities is explained; it involves manipulating one side of the equation to transform it into the other side. The text emphasizes that trigonometric identities are not treated like standard equations; operations are not performed on both sides simultaneously. Instead, the goal is to transform one expression into another, changing its appearance without altering its value. This is a crucial skill in simplifying and solving more complex trigonometric problems. The text highlights the use of writing everything in terms of sine and cosine as a common starting point to verify the identities, often utilizing tools like common denominators and the fundamental Pythagorean identity (sin²θ + cos²θ = 1).

This section introduces double-angle identities—relationships between trigonometric functions of an angle and those of double the angle (e.g., 2θ). The derivation of these formulas is discussed, noting the manipulation required to obtain the desired expressions, which often involves employing identities for negative angles, leveraging the understanding of sine and cosine's behavior in different quadrants. The text emphasizes that these identities are invaluable tools for simplifying expressions and solving more complex trigonometric problems, particularly when dealing with equations involving angles that are multiples of a given angle.

The section transitions to solving trigonometric equations—equations that are only true for specific angle values. The text highlights that these equations often have multiple solutions due to the cyclical nature of trigonometric functions. The process of solving such equations is illustrated with examples, using techniques that may involve employing technology (calculators) or applying knowledge of reference angles and the signs of trigonometric functions in different quadrants. It's noted that some equations may have no real solutions—for example, sin x = 2 has no real solution because the sine function's range is between -1 and 1. The text demonstrates how to find solutions within the range of 0 to 360 degrees, considering the quadrantal angles and potential for multiple solutions within a single period of a trigonometric function.

This section introduces the Law of Sines, a fundamental tool for solving triangles that are not right-angled. The Law of Sines establishes a relationship between the angles and the lengths of the sides of any triangle. It states that the ratio of the length of a side to the sine of the opposite angle is constant for all three sides of the triangle. The text notes that this law is particularly useful when you know two angles and one side of a triangle, or one angle and two sides (provided one side corresponds to the known angle). The text however also highlights the 'ambiguous case' associated with the Law of Sines, where using this law to find an angle can lead to multiple solutions. This ambiguity arises because the sine function has positive values in both the first and second quadrants; therefore, there might be two possible angles for a given ratio. There are various scenarios that can be found in the ambiguous case.

The ambiguous case of the Law of Sines is discussed in detail, explaining that when using the law with one angle and two sides, there might be one solution, two solutions, or no solution for the triangle. Six different scenarios are briefly mentioned, leading to either one, two, or zero triangles. This ambiguity is a direct consequence of the inverse trigonometric functions which can produce multiple angles for a given trigonometric value. The section clearly describes that the ambiguity arises because the sine function can have the same value for two different angles. The text explains that this ambiguity requires a careful consideration of the possible solutions when applying the Law of Sines.

The Law of Cosines is introduced as an alternative method for solving triangles, particularly useful in situations where the Law of Sines is insufficient or ambiguous. Unlike the Law of Sines, the Law of Cosines is applicable when three sides of a triangle are known, allowing the calculation of the angles, or when two sides and the included angle are known, to calculate the third side. The text mentions that using the Law of Cosines is computationally more involved than using the Law of Sines. It is often a necessary tool when an angle-side pair is not available to apply the Law of Sines. Several examples of using the Law of Cosines are given, involving calculations related to distances between locations given bearings and speeds. These applications demonstrate the practicality of the Law of Cosines for solving real-world problems related to navigation and distance calculations.