Trigonometric Tables
Trigonometric tables provide the numerical values for the fundamental trigonometric functions of angles, such as the cosine of 30° or the sine of 60°. The tables are usually arranged with the angles listed in degrees (or sexagesimal numeration) and radians. The intersection of a function name with the angle provides the numerical value.
Many sources provide a trigonometric table for every whole number angle from 0 to 90° (or 0 to /2 radians). In these tables the angles from 0 to 45° are listed in the left-hand column and are used with the functions listed across the top row. The angles from 45 to 90° are listed in the right-hand column in descending order and are used with the functions listed across the bottom row. Here is an excerpt from such a table:
From this table, the sine of 30° is 0.5000. Using the functions listed across the bottom row, the cosine of 60° also is 0.5000. In fact, this table arrangement is possible because of the mathematical relationship between certain functions for complementary angles (i.e., two angles that sum to 90°). Thus, tan 1° = cot 89°, sec 45° = csc 45°, etc.
Note that cot 0°, tan 90°, csc 0°, and sec 90° are not defined. This is because there are certain angles for which the trigonometric functions do not exist.
This same type of table can be used to find the trigonometric functions of any angle Θ with the help of a reference angle and the proper sign conventions.
Reference angles are calculated as follows:
- For 0°<Θ<90°, Reference angle = Θ
- For 90°<Θ<180°, Reference angle = 180°-Θ
- For 180°<Θ<270°, Reference angle = Θ-180°
- For 270°<Θ<360°, Reference angle = 360°-Θ
The numerical value of a trigonometric function of angle Θ is equivalent to the absolute value of the same function of the reference angle. The sign of the numerical value is determined by the quadrant in which the terminal side of Θ falls.
For example, to find cos 210°, the reference angle = 210°-180° = 30°. From the table, cos 30° = 0.8660. Because the terminal side of a 210° angle falls in the 3rd quadrant, the sign is negative. Therefore, cos 210° = -.8660.
Likewise, to find sec 315°, the reference angle = 360°-315° = 45°. From the table, sec 45° = 1.4142. Because the terminal side of a 315° angle falls in the 4th quadrant, the sign is positive. Therefore, sec 315° = 1.4142.
For negative angles (i.e., those that rotate clockwise, instead of counterclockwise), use the absolute value of the desired angle Θ to determine the reference angle. For example, to determine cos -120°, use Θ = 120°. The reference angle = 180°-120° = 60°. From the table, cos 60° = 0.5000. The terminal side of a -120° angle falls in the 3rd quadrant, therefore, cos -120° = -0.5000.
For angles greater than 360°, subtract multiples of 360° until Θ is less than 360°. For example, cos 765° is the same as cos 765°-360°-360° or cos 45°, which is 0.7071.
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