Pre-Calc Exam Notes 25

Pre-Calc Exam Notes 25 - Trigonometric Functions of Any...

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Unformatted text preview: Trigonometric Functions of Any Angle • Section 1.4 25 We can now define the trigonometric functions of any angle in terms of Cartesian coordinates. Recall that the x y-coordinate plane consists of points denoted by pairs ( x, y) of real numbers. The first number, x, is the point’s x coordinate, and the second number, y, is its y coordinate. The x and y coordinates are measured by their positions along the x-axis and y-axis, respectively, which determine the point’s position in the plane. This divides the x y-coordinate plane into four quadrants (denoted by QI, QII, QIII, QIV), based on the signs of x and y (see Figure 1.4.3(a)-(b)). y QII x<0 y>0 QIII x<0 y<0 y QI x>0 y>0 0 (2, 3) (−3, 2) x x 0 QIV x>0 y<0 (−2, −2) (a) Quadrants I-IV (3, −3) (b) Points in the plane y ( x , y) r θ x 0 (c) Angle θ in the plane Figure 1.4.3 x y-coordinate plane Now let θ be any angle. We say that θ is in standard position if its initial side is the positive x-axis and its vertex is the origin (0, 0). Pick any point ( x, y) on the terminal side of θ a distance r > 0 from the origin (see Figure 1.4.3(c)). (Note that r = x2 + y2 . Why?) We then define the trigonometric functions of θ as follows: sin θ = y r cos θ = x r tan θ = y x (1.2) csc θ = r y sec θ = r x cot θ = x y (1.3) As in the acute case, by the use of similar triangles these definitions are well-defined (i.e. they do not depend on which point ( x, y) we choose on the terminal side of θ ). Also, notice that | sin θ | ≤ 1 and | cos θ | ≤ 1, since | y | ≤ r and | x | ≤ r in the above definitions. ...
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This note was uploaded on 01/21/2012 for the course MAC 1130 taught by Professor Dr.cheun during the Fall '11 term at FSU.

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