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Chapter 1.6 Trigonometric Functions

This chapter covers the fundamentals of trigonometric functions, focusing on the unit circle and converting between degrees and radians. You will learn how to convert angles using the formulas ( text{radians} = frac{pi}{180}times text{degrees} ) and ( text{degrees} = frac{180}{pi}times text{radians} ). We explore the definition of radians in terms of arc length and how to find the lengths of arcs for given angles. Additionally, we delve into the six trigonometric functions and transformations of their graphs, as well as how to use inverse functions to determine angle measures.

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Chapter 1.6 Trigonometric Functions

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  1. Chapter 1.6 Trigonometric Functions

  2. The Unit Circle

  3. Degree/Radian Conversion • To convert a degree measure to radians, multiply by π radians180° • To convert a radian measure to degrees, multiply by 180°π radians

  4. Examples • Examples • 1) 120° • 2) -45° • 3) 5π6 • 4) -3π2

  5. Radian Measure • The RADIAN MEASURE of the angle ACB at the center of the unit circle equals the length of the arc that ACB cuts from the unit circle. Radius =1

  6. Finding Arc Length • Find the length of an arc on a circle of radius 3 by a central angle of measure 2π/3. • S = r θ • = 3(2π/3) • = 2π

  7. An Angle θ In Standard Position When an angle of measure θ is placed in standard position at the center of a circle of radius r, the six trigonometric functions of θ are defined as follows: sin θ = y/r cscθ = r/y Cos θ = x/r sec θ = r/x Tan θ = y/x cot θ = x/y

  8. (SOHCAHTOA) • Sin – opp/hyp • Cos – adj/hyp • Tan – opp/adj • Csc – hyp/opp • Sec – hyp/adj • Cot – adj/opp

  9. Graph of sin

  10. Graph of cos

  11. Graph of tan

  12. Periodicity • Periodic Function, Period: A function f(x) is periodic if there is a postive number p such that f(x + p) = f(x) for every value of x. The smallest such value of p is the period of f.

  13. Transformations of Trigonometric Graphs • Y = a f ( b ( x + c ) ) + d • A = vertical stretch or shrink/reflection about x-axis • B = horizontal stretch or shrink/ reflection about y-axis • C = Horizontal shift • D = vertical shift

  14. Finding Angles in degrees and Radians • Find the measure of cos-1 (-0.5) in degrees and radians. • Put the calculator in degree mode and enter cos-1 (-0.5). You will get 120 degrees.

  15. Using the Inverse Trigonometric Functions • Sinx = 0.7 • Take the sin-1 of both sides. • X = sin-1(0.7) • X = 0.775

  16. Homework • Quick Review pg 52 # 1-4 • Section 1.6 Exercises pg 52 #1-10

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