Technical Mathematics, Sixth Edition

7.4. An Angle in Standard Position

Early in this chapter we defined the trigonometric funcions in terms of the sides of a right triangle. For many purposes, such as the vectors we will work with in the next section, it is more useful to define the trig functions in terms of an angle drawn on coordinate axes, as shown in Fig. 7-38. When drawn in this way, with the vertex at the origin O and with one side along the x axis, the angle is said to be in standard position. We may think of an angle as being generated by a line rotating counterclockwise from an initial position on the x axis to some terminal position.

Next we select any point P on the terminal side of the angle, with rectangular coordinates x and y. We form a right triangle OPQ by dropping a perpendicular from P to the x axis. This side OQ is adjacent to angle θ and has a length x. Similarly side PQ is opposite to angle θ and has a length y. The side OP is the hypotenuse of the right triangle, and we label its length r. Note that by the Pythagoran theorem, r² = x² + y².

As before, we define the sine of θ as the ratio of opposite to hypotenuse. But here it is also the ratio of the distance y to the distance r. Similarly the cosine and tangent can be defined in terms of x, y, and r. Our three trignometric functions, defined both as ratios of the sides of a right triangle and as the coordinates of a point on the terminal side of an angle in standard position, are then

Figure 7.38. An angle in standard position.

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