Trigonometry

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

2. Chapter 2: Trigonometric Functions 2.1 Angles and Arcs 2.2 Right Triangle Trigonometry 2.3 Trigonometric Functions of Any Angle 2.4 Trigonometric Functions of Real Numbers 2.5 Graphs of the Sine and Cosine Functions 2.6 Graphs of the other Trigonometric Functions 2.7 Graphing Techniques 2.8 Harmonic Motion – An Application of the Sine and Cosine Functions

3. Chapter 2 Overview • Chapter 2 begins by dealing with the fundamentals of angular measurement and the relationships between angular and linear quantities. Trigonometric values are then defined first as ratios of sides in right triangles, then as ratios of coordinates on the terminal side of any angle and finally as coordinates of points on the unit circle. Even though a precise definition is given for each trigonometric function no method of generally calculating their values is given (until Calculus 3). Thus trigonometry becomes the study of the properties of the functions themselves and evaluation is left to computing machines. The transformation and graphing of trigonometric functions is covered in the last half of chapter 2.

4. 2.1: Angles and Arcs 1 • Degree Measure • Complementary and Supplementary Angles. Example 1 • Coterminal Angles. Example 2 • Radian: The angle subtended by one radius of arc. • Degrees ↔ Radian Conversions. Examples 3 & 4 • Revolutions ↔ Radian Conversions. Example 7 • Linear and Angular Relationships Examples 5, 6, & 8

5. 2.2: Right Angle Trigonometry 1 • Right Triangle Trigonometric definitions: Examples 1, 2, 4, 5 & 6

6. 2.2: Right Angle Trigonometry 2 • Reference Triangles Example 3

7. 2.3: Trigonometric Functions of Any Angle 1 • Any Angle Trigonometric definitions: Example 1 • Quadrantal angles and Quadrant signs. Example 2 • Reference angles. Examples 3 & 4

8. 2.4: Trigonometric Functions of Real Numbers 1 • The Wrapping function maps Arc Lengths = Angles (real numbers) to points (x=cos[t], y=sin[t]) on the Unit Circle. Example 1 • Real Number Trigonometric definitions: Examples 2 & 3

9. 2.4: Trigonometric Functions of Real Numbers 2 • The Domain and Range of the Trigonometric Functions. • Even and Odd Trigonometric Functions. Example 4 • Periods of the Trigonometric Functions. • Trigonometric Identities. Examples 5-7

10. 2.5: Graphs of the Sine and Cosine Functions 1 • Transforming a function: Class Example • a is the vertical scaling factor. • w is the horizontal scaling factor. • h is the horizontal shift. • k is the vertical shift.

11. 2.5: Graphs of the Sine and Cosine Functions 2 • Sine and Cosine Curves: Periods, Quadrants, Quadrantal Angles and Reference Angles.

12. 2.5: Graphs of the Sine and Cosine Functions 3 • Transforming a periodic function: • A is the amplitude. • ω is the angular frequency. • h is the phase shift. • k is the vertical offset. • The period of the function is its normal period divided by the angular frequency.

13. 2.5: Graphs of the Sine and Cosine Functions 4 • Graphing a Trigonometric Function. Examples 1-7 • Write the equation in Standard Form and identify A, ω, h and k. • Compute the period and quarter period. • Draw the envelope (if possible) and center lines. • Find a divisor compatible (worst case is 1) with both the quarter period and the horizontal offset. Mark the x-axis and center line in these units. • Mark the horizontal offset point and quarter periods from that point in both directions on the center line. Draw in any vertical asymptotes. • For each quarter period point mark the appropriate height of function on the graph. When graphing the cotangent, secant and cosecant functions use the reciprocal of values from the tangent, cosine and sine functions. • Connect the marks.

14. 2.6: Graphs of the Other Trigonometric Functions 1 • Tangent and Cotangent Curves: Periods, Quadrants, Quadrantal Angles and Reference Angles.

15. 2.6: Graphs of the Other Trigonometric Functions 2 • Secant and Cosecant Curves: Periods, Quadrants, Quadrantal Angles and Reference Angles.

16. 2.6: Graphs of the Other Trigonometric Functions 3 • Graphing a Trigonometric Function. Examples 1-5 • Write the equation in Standard Form and identify A, ω, h and k. • Compute the period and quarter period. • Draw the envelope (if possible) and center lines. • Find a divisor compatible (worst case is 1) with both the quarter period and the horizontal offset. Mark the x-axis and center line in these units. • Mark the horizontal offset point and quarter periods from that point in both directions on the center line. Draw in any vertical asymptotes. • For each quarter period point mark the appropriate height of function on the graph. When graphing the cotangent, secant and cosecant functions use the reciprocal of values from the tangent, cosine and sine functions. • Connect the marks.

17. 2.7: Graphing Techniques 1 • Graphing a Trigonometric Function. Examples 1-6 • Write the equation in Standard Form and identify A, ω, h and k. • Compute the period and quarter period. • Draw the envelope (if possible) and center lines. • Find a divisor compatible (worst case is 1) with both the quarter period and the horizontal offset. Mark the x-axis and center line in these units. • Mark the horizontal offset point and quarter periods from that point in both directions on the center line. Draw in any vertical asymptotes. • For each quarter period point mark the appropriate height of function on the graph. When graphing the cotangent, secant and cosecant functions use the reciprocal of values from the tangent, cosine and sine functions. • Connect the marks.

18. 2.7: Graphing Techniques 2 • Addition of Ordinates (calculator). Examples 7 & 8 • Damping Factors (calculator). Example 9

19. 2.8: Harmonic Motion 1 • If the sinusoidal curve (sine or cosine) is shifted so that it begins at the origin and is centered on the x-axis then become • The frequency of a sinusoidal function is defined to be the reciprocal of the period. The phase shift is then and the sinusoidal equations reduce to Example 1

20. 2.8: Harmonic Motion 2 • Simple Harmonic Motion: periodic motion induced by a restoring force that is directly proportional to the displacement from the rest position. For springs this is mathematically represented by Hooke’s Law: F = - k x. • Physics shows that the frequency of oscillation for a mass-spring system is Example 2 • Damped Harmonic Motion. Example 3 • Mass-Spring Experiment.