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Chapter 4

Chapter 4 . 4.2 Trigonometric functions: The unit circle. Objectives. Identify a unit circle Evaluate trigonometric functions Use domain and period to evaluate sine and cosine functions. Unit circle.

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Chapter 4

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  1. Chapter 4 4.2 Trigonometric functions: The unit circle

  2. Objectives • Identify a unit circle • Evaluate trigonometric functions • Use domain and period to evaluate sine and cosine functions

  3. Unit circle • There are six basic trigonometric functions that we will use over and over throughout the course.  There are two ways that we can define these functions, each having it's own advantages and disadvantages.  The first way that we will look at the trig functions is as points on the unit circle. 

  4. Unit Circle • First, we need to understand the unit circle.  As the name  implies, it is a circle where the radius is 1 (one unit).  The unit circle equation is x² + y² = 1 and the graph looks like this: 

  5. Unit circle • Now, if we start at the point (1, 0) and walk a distance t around the circle, we will arrive at a point (x, y) represented by the blue point on the circle in the figure below.  The distance traveled, t, is shown in red.  At the right of the circle is a red line segment that ends in a blue point that is exactly the same length as the red arc ending in the blue point. We are interested in the (x, y) coordinates of the point corresponding to going around the circle a distance of t. 

  6. Trigonometric functions • There are six trigonometric functions: sine (abbreviated sin), cosine (abbreviated cos), tangent (abbreviated tan), cosecant (abbreviated csc), secant (abbreviated sec), and cotangent (abbreviated cot).   • Here's how the trigonometric functions are defined.  Let t be the distance traveled around the unit circle ending at the point (x, y):sin(t) = y,cos(t) = x,tan(t) = y/x so long as x is not 0,csc(t) = 1/y, so long as y is not 0,sec(t) = 1/x so long as x is not 0,cot(t) = x/y so long as y is not 0.Note that csc is the reciprocal of sin, sec is the reciprocal of cos, and cot is the reciprocal of tan

  7. Visualizing trigonometric functions • There are some important angles to know by heart (memorize).  They are summarized in the figure below. Here's how to read the figure.  Look at the point corresponding to t = π/3.  The x-coordinate is 1/2 = 0.5 and the y-coordinate is sqrt(3)/2.  Therefore cos(π/3) = 0.5 and the sin(π/3) = sqrt(3)/2.

  8. Unit circle

  9. Evaluating trigonometric functions • Evaluate the six trigonometric functions at each real number • A) t= • B) • C)

  10. Student guided practice • Do problems 23-26 in your book page 270

  11. Domain and period of sine and cosine • If an angle corresponds to a point Q(x,y) on the unit circle, it is not hard to see that the angle corresponds to the same point Q(x,y), and hence that • Moreover, is the smallest positive angle for which Equations 1 are true for any angle . In general, we have for all angles : • We call the number the period of the trigonometric functions and , and refer to these functions as being periodic. Both and are periodic functions as well, with period , while and are periodic with period .

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