# 1: Right Triangle Trigonometry Angles

- Page ID
- 3216

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

Trigonometry is the study of the relations between the sides and angles of triangles. The word “trigonometry” is derived from the Greek words trigono (τρ´ιγωνo), meaning “triangle”, and metro (µǫτρω´ ), meaning “measure”. Though the ancient Greeks, such as Hipparchus and Ptolemy, used trigonometry in their study of astronomy between roughly 150 B.C. - A.D. 200, its history is much older. For example, the Egyptian scribe Ahmes recorded some rudimentary trigonometric calculations (concerning ratios of sides of pyramids) in the famous Rhind Papyrus sometime around 1650 B.C. Trigonometry is distinguished from elementary geometry in part by its extensive use of certain functions of angles, known as the trigonometric functions. Before discussing those functions, we will review some basic terminology about angles.

- 1.1: Angles
- In elementary geometry, angles are always considered to be positive and not larger than \(360^\circ \). You also learned that the sum of the angles in a triangle equals \(180^◦\), and that an isosceles triangle is a triangle with two sides of equal length. Recall that in a right triangle one of the angles is a right angle. Thus, in a right triangle one of the angles is \(90^◦\) and the other two angles are acute angles whose sum is \(90^◦\) (i.e. the other two angles are complementary angles).

- 1.2: Trigonometric Functions of an Acute Angle
- For a right triangle △ABC, with the right angle at C and with lengths a, b, and c. For the acute angle A, call the leg BC its opposite side, and call the leg AC its adjacent side. Recall that the hypotenuse of the triangle is the side AB. The ratios of sides of a right triangle occur often enough in practical applications to warrant their own names, so we can define the six trigonometric functions of A.

- 1.3: Applications and Solving Right Triangles
- Throughout its early development, trigonometry was often used as a means of indirect measurement, e.g. determining large distances or lengths by using measurements of angles and small, known distances. Today, trigonometry is widely used in physics, astronomy, engineering, navigation, surveying, and various fields of mathematics and other disciplines. In this section we will see some of the ways in which trigonometry can be applied. Your calculator should be in degree mode for these examples.

- 1.4: Trigonometric Functions of Any Angle
- To define the trigonometric functions of any angle - including angles less than 0° or greater than 360° - we need a more general definition of an angle. We say that an angle is formed by rotating a ray OA about the endpoint O (called the vertex), so that the ray is in a new position, denoted by the ray OB. The ray OA is called the initial side of the angle, and OB is the terminal side of the angle.

- 1.5: Rotations and Reflections of Angles
- Now that we know how to deal with angles of any measure, we will take a look at how certain geometric operations can help simplify the use of trigonometric functions of any angle, and how some basic relations between those functions can be made. The two operations on which we will concentrate in this section are rotation and reflection.

- 1.E: Right Triangle Trigonometry Angles (Exercises)
- These are homework exercises to accompany Corral's "Elementary Trigonometry" Textmap. This is a text on elementary trigonometry, designed for students who have completed courses in high-school algebra and geometry. Though designed for college students, it could also be used in high schools. The traditional topics are covered, but a more geometrical approach is taken than usual. Also, some numerical methods (e.g. the secant method for solving trigonometric equations) are discussed.

*Thumbnails: Types of angles.*

### Contributors

Michael Corral (Schoolcraft College). The content of this page is distributed under the terms of the GNU Free Documentation License, Version 1.2.