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Trigonometry

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.

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Trigonometry

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

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

  3. 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. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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.

  10. 4.3: Vectors 5 • Work is the product of force in the direction of motion and the distance moved. Example 11

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