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Trigonometry: Graphs

Graphs of Trigonometric Functions

Problems

Problems

Below are the graphs of the six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. On the $x$-axis are values of the angle in radians, and on the $y$-axis is f (x) , the value of the function at each given angle.

Figure %: Graphs of the six trigonometric functions

Convince yourself that the graphs of the functions are correct. See that the signs of the functions do indeed correctly correspond with the signs diagrammed in the in Trigonometric Functions, and that the quadrantal angles follow the rules described in the .

Also, for example, consider the definition of sine. Given a point on the terminal side of an angle, the sine of the angle is the ratio of the y-coordinate of that point to the distance between it and the origin. Now imagine that angle changing, but the point remaining the same distance from the origin. The point traces the circumference of a circle. As the angle goes from 0 to radians, the y coordinate increases, and so does the sine of the angle. As the angle goes from radians to Π radians, the y-coordinate decreases, and so does the sine of the angle, but each is still positive. Then as the terminal side of the angle enters the third and fourth quadrant, the y coordinate of the point on the terminal side is negative, and first decreases, and then increases. All of these changes are indeed reflected in the graph. Below in the figure the quadrants of the coordinate plane are represented in the graph.

Figure %: The quadrants are represented in graphs as well as the coordinate plane

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