Trigonometry Cloud County Community College Spring, 2012 Instructor: Timothy L. Warkentin

Chapter 4: Applications of Trigonometry 4.1 The Law of Sines 4.2 The Law of Cosines and Area 4.3 Vectors

Chapter 4 Overview Trigonometry may be the most useful and practical mathematical tool ever discovered. This chapter shows how Trigonometry can be applied to a wide range of common situations including the solution of oblique triangles and vector problems.

4.1: The Law of Sines 1 Solving a triangle means finding the three missing elements (angles or sides) of the six possible elements. The five solvable triangles are: AAS, ASA, ASS, SAS, & (SSS – Law of Cosines needed). AAA is not solvable. The Law of Sines: The Law of Sines can yield indefinite solutions for angles as a result of the sine function being positive in the first two quadrants (the quadrants involved in oblique triangles). Solving triangles with the Law of Sines. Examples 1, 2 & 4 Solving the Ambiguous Triangle Case (ASS) using the Law of Sines. Example 2

4.2: The Law of Cosines and Area 1 The Law of Cosines: The Law of Cosines yields definite solutions for angles because the cosine function has both positive and negative values in the first two quadrants (the quadrants involved in oblique triangles). Solving triangles with the Law of Cosines: Examples 1-3 Finding the area of a triangle: Examples 4 & 5 Finding the area of a triangle with Heron’s Formula: Examples 6 & 7

4.3: Vectors 1 Scalar: a number indicating the magnitude of a measurement. Vector: a number indicating the magnitude and direction of a measurement. Vectors can be represented by explicitly giving the magnitude and direction (polar form, ), by an arrow diagram, the & unit vectors (preferred in physics) or by an ordered pair (preferred in mathematics). When a vector is written as an ordered pair the coordinates are called the vector’s components. Example 1

4.3: Vectors 2 The sum/difference of two vectors is called the Resultant. The resultant may be approximated graphically or found exactly by adding/subtracting the ordered pairs (components). The Norm or Magnitude of a vector is found by using the Pythagorean Theorem with the components. A vector may be scaled (Scalar Multiplication) by multiplying the vector’s components by the scalar. Example 2 The Unit Vector for a vector is a vector with norm equal to one and in the same direction. Example 3

4.3: Vectors 3 Physicists commonly write vectors in the form where and are unit vectors in the horizontal and vertical directions respectively. Example 4 Finding the components of a vector. Examples 5 – 7 The Dot Product (a scalar) of two vectors: Example 8

4.3: Vectors 4 Finding the Angle Between Two Vectors. Example 9 The Projection of onto (a length): Example 10 Vectors are parallel when the angle between them is 0 or 180 degrees (one is a scalar multiple of the other). Vectors are perpendicular (orthogonal) when the angle between the them is 90 degrees.

4.3: Vectors 5 Work is the product of force in the direction of motion and the distance moved. Example 11