By Clark G. L.

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Figure 2-3: Vertical and horizontal distances between points. Finding the distance between pairs of vertical or horizontal points, (x1,y1) and (x2,y2), is easy: ✓ Vertical distance (the x-coordinates are the same) is |y1 – y2|. ✓ Horizontal distance (the y-coordinates are the same) is |x1 – x2|. 33 34 Part I: Getting Started with Trigonometry Another slant: Diagonal distances Sometimes the distances or lengths you want to determine are on a slant — they go diagonally from one point to another. The formula for determining these distances is based on the Pythagorean theorem.

Some angles and trig functions have nice values, but most don’t. Table 1-2 shows the trig functions for a 30-degree angle. 000 Some characteristics that the entries in Table 1-2 confirm are that the sine and cosine functions always have values that are between and including –1 and 1. Also, the secant and cosecant functions always have values that are equal to or greater than 1 or equal to or less than –1. ) Using the table in the Appendix, you can find more values of trig functions for particular angle measures (in degrees): tan 45° = 1 csc 90° = 1 sec 60° = 2 I chose these sample values so the answers look nice and whole.

The –3 is smaller, so subtracting in that order gave a positive answer. The alternative is to subtract in the opposite order and take the absolute value of the result. If you do –3 – 6, you get –9. The absolute value of –9, written |–9|, equals 9. Horizontal distances work the same way. In Figure 2-3, you can see the horizontal distance between two points. To find the distance between the points (–8,2) and (5,2), just calculate the difference between –8 and 5, because the y-coordinates are the same.

### Correctness of the definition of a PD operator by Clark G. L.

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