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Chapter 1 | Functions and Graphs
Figure 1.31 The angle θ is in standard position. The values of the trigonometric functions for θ are defined in terms of the coordinates x and y .
Definition Let P = ( x , y ) be a point on the unit circle centered at the origin O . Let θ be an angle with an initial side along the positive x -axis and a terminal side given by the line segment OP . The trigonometric functions are then defined as (1.9) sin θ = y csc θ = 1 y cos θ = x sec θ = 1 x tan θ = cot θ = x y If x =0, sec θ and tan θ are undefined. If y =0, then cot θ and csc θ are undefined. We can see that for a point P = ( x , y ) on a circle of radius r with a corresponding angle θ , the coordinates x and y satisfy cos θ = x r x = r cos θ sin θ = y r y = r sin θ . The values of the other trigonometric functions can be expressed in terms of x , y , and r ( Figure 1.32 ). y x
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