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Consider, for example, the fact that tan (p /4) = 1 A true function that “undoes” the tangent must map an input value of 1 in the domain to an output value of p /4 in the range, but to no other values. In fact, no matter what we input to the function, we must never get more than one output. To ensure that the inverse of the tangent behaves as a true function, we must restrict its range (output) to an open interval where we don’t get any redundancy. By convention, mathematicians specify the open interval (−p /2,p /2) for this purpose.

Imagine that you are in the middle of a flat, open field at night. The ball describes a circle as viewed from high above, as shown in Fig. 2-2A. If a friend stands far away with her eyes exactly in the plane of the ball’s orbit, she sees a point of light that oscillates back and forth, from right-to-left and left-to-right, along what appears to be a straight-line path (Fig. 2-2B). Starting from its rightmost apparent position, the glowing point moves toward the left for 1/2 second, speeding up and then slowing down; then it reverses direction; then it moves toward the right for You Ball A Top view String Ball B Side view Figure 2-2 Orbiting ball and string.

Angle variables can also be represented by more familiar characters such as x, y, or z. As long as we know the context and stay consistent in a given situation, it really doesn’t matter what we call an angle. Radian measure Imagine two rays pointing outward from the center of a circle. Each ray intersects the circle at a point. Suppose that the distance between these points, as measured along the arc of the circle, is equal to the radius of the circle. In that case, the measure of the angle between the rays is one radian (1 rad).