Chapter 3: Right Triangle Trigonometry

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Page ID: 145904

Roy Simpson & Katherine Yoshiwara
Los Angeles Pierce College

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How would one measure the distance to an inaccessible object, such as a ship at sea? In the 6^th century BC, the Greek philosopher Thales estimated the distances to ships at sea using triangulation, a method for calculating distances by forming triangles. Using Trigonometry and the measured length of just one side, the lengths of the other sides can be calculated.

Triangulation has been used to compute distances ever since. In the 16^th century, mapmakers used triangulation to position distant places accurately. As new methods in navigation and astronomy required greater precision, the idea of a survey using chains of triangles was developed.

In 1802, the East India Company embarked on the Great Trigonometrical Survey of India, which aimed to measure the entire Indian subcontinent with scientific precision.

$1922-index-of-great-trigonometrical-survey-of-india-KBTD0G.jpg$

The surveyors began by measuring a baseline near Madras. The baseline was the only distance they measured; all other distances were calculated from it using measured angles. Each calculated distance became the base side of another triangle used to calculate the distance to another point, which in turn started another triangle. Eventually this process formed a chain of triangles connecting the origin point to other locations.

Because of the size of the area to be surveyed, the surveyors did not triangulate the whole of India. Instead, they created a ”gridiron” of triangulation chains running from north to south and east to west, as seen on the survey map above.

The survey, completed in 1871, calculated the heights of the Himalayan giants: Everest, K2, and Kanchenjunga. It also provided one of the first accurate measurements of a section of an arc of longitude.

Today, triangulation is used for many purposes, including surveying, navigation, metrology, astrometry, binocular vision, and location of earthquakes.

3.1: The Trigonometric Functions - Right Triangle Definition
This section defines trigonometric functions based on right triangle ratios, covering sine, cosine, and tangent alongside their reciprocals. It explores calculating trigonometric values for angles within right triangles, emphasizes the significance of the Pythagorean Theorem in finding these values, and introduces the concept of cofunctions. Through examples and checkpoints, it lays the groundwork for understanding trigonometric relationships and their applications beyond the coordinate system.
3.2: Technology and Right Triangle Trigonometry
This section explores how to use technology to work with Right Triangle Trigonometry. It covers computing values of trigonometric functions, converting between degrees, minutes, and seconds (DMS) and decimal degrees (DD), and finding acute angles using inverse trigonometric functions. Practical examples and checkpoints help readers learn to navigate between different angle measurements and utilize technology effectively for trigonometric calculations.
3.3: Solving Right Triangles
This section is centered on solving right triangles, a fundamental concept in Trigonometry that involves finding the missing sides and angles of a right triangle when given some of these measurements. It explains how to use trigonometric ratios—sine, cosine, and tangent—to find unknown sides or angles, illustrating this with examples. Additionally, the text discusses the application of trigonometry in real-world problems, such as determining distances or heights that are not directly measurable.
3.4: Applications Involving Right Triangles
This section dives into real-world applications of right triangle trigonometry, focusing on applications involving angles of elevation and depression as well as navigation. It covers how to measure and use angles to calculate distances and heights in practical scenarios, such as determining the height of a cliff or navigating using bearings or headings. This foundational knowledge is crucial for applying Trigonometry to solve practical problems encountered in various fields.

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