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3: Triangles and Vectors
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- 3.1: Trigonometric Functions of Angles
- As was stated at the start of Chapter 1, trigonometry had its origins in the study of triangles. In fact, the word trigonometry comes from the Greek words for triangle measurement. We will see that we can use the trigonometric functions to help determine lengths of sides of triangles or the measure on angles in triangles. As we will see in the last two sections of this chapter, triangle trigonometry is also useful in the study of vectors.
- 3.2: Right Triangles
- In this section, we will learn how to use the trigonometric functions to relate lengths of sides to angles in right triangles and solve this problem as well as many others.
- 3.3: Triangles that are Not Right Triangles
- There are many triangles without right angles (these triangles are called oblique triangles). Our next task is to develop methods to relate sides and angles of oblique triangles. In this section, we will develop two such methods, the Law of Sines and the Law of Cosines. In the next section, we will learn how to use these methods in applications.
- 3.4: Applications of Triangle Trigonometry
- It should then be no surprise that we can use the Law of Sines and the Law of Cosines to solve applied problems involving triangles that are not right triangles. In most problems, we will first get a rough diagram or picture showing the triangle or triangles involved in the problem. We then need to label the known quantities. Once that is done, we can see if there is enough information to use the Law of Sines or the Law of Cosines.
- 3.5: Vectors from a Geometric Point of View
- There are some quantities that require only a number to describe them. We call this number the magnitude of the quantity. One such example is temperature since we describe this with only a number such as 68 degrees Fahrenheit. Other such quantities are length, area, and mass. These types of quantities are often called scalar quantities. However, there are other quantities that require both a magnitude and a direction. One such example is force, and another is velocity.
- 3.6: Vectors from an Algebraic Point of View
- We have seen that a vector is completely determined by magnitude and direction. So two vectors that have the same magnitude and direction are equal. That means that we can position our vector in the plane and identify it in different ways. Vectors also have certain geometric properties such as length and a direction angle. With the use of the component form of a vector, we can write algebraic formulas for these properties.
- 3.E: Triangles and Vectors (Exercises)