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Pre-Calc Exam Notes 103

Pre-Calc Exam Notes 103 - circle its y-coordinate is sin θ...

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5 Graphing and Inverse Functions The trigonometric functions can be graphed just like any other function, as we will now show. In the graphs we will always use radians for the angle measure. 5.1 Graphing the Trigonometric Functions x y s = r θ = θ 1 θ 1 0 ( x , y ) = (cos θ , sin θ ) x 2 + y 2 = 1 Figure 5.1.1 The first function we will graph is the sine func- tion. We will describe a geometrical way to create the graph, using the unit circle . This is the circle of radius 1 in the xy -plane consisting of all points ( x , y ) which satisfy the equation x 2 + y 2 = 1. We see in Figure 5.1.1 that any point on the unit circle has coordinates ( x , y ) = (cos θ , sin θ ), where θ is the angle that the line segment from the origin to ( x , y ) makes with the positive x -axis (by definition of sine and cosine). So as the point ( x , y ) goes around the
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Unformatted text preview: circle, its y-coordinate is sin θ . We thus get a correspondence between the y-coordinates of points on the unit circle and the values f ( θ ) = sin θ , as shown by the horizontal lines from the unit circle to the graph of f ( θ ) = sin θ in ±igure 5.1.2 for the angles θ = 0, π 6 , π 3 , π 2 . θ f ( θ ) 1 π 6 π 3 π 2 2 π 3 5 π 6 π f ( θ ) = sin θ π 6 π 3 π 2 1 1 x 2 + y 2 = 1 θ Figure 5.1.2 Graph of sine function based on y-coordinate of points on unit circle We can extend the above picture to include angles from 0 to 2 π radians, as in ±igure 5.1.3. This illustrates what is sometimes called the unit circle deFnition of the sine function . 103...
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