TRIGONOMETRIC FUNCTIONS Machinery's Handbook, 31st Edition
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Fig. 3. Signs of the Three Main Trigonometric Functions Fig. 3 shows the values of angles and ratios on the unit circle in great detail, as well as the signs of the functions in each of the four quadrants. Radian and degree measure are also marked off. The figure also shows the sign (+ or –) and the values of trigonometric functions for angles in each of the four quadrants, 0 to 90, 90 to 180, 180 to 270, and 270 to 360 degrees. A π radian is approximately 3.14, and 2 π is approximately 6.3 radian measure, as the completed circle indicates. The corresponding degree measures are marked as well, as are the pure radian equivalents. The chart indicates, for example, that all the functions are positive for angles between 0 and 90 degrees. This is because x and y are both positive in the first quadrant, and all the trigonometric ratios are therefore also positive. In the same way, the cotangent of an angle between 180 and 270 degrees is positive and has a value between infinity and 0; in other words, the cotangent for 180 degrees is infinitely large and then the cotangent gradually decreases for increasing angles, so that the cotangent for 270 degrees equals 0. The cosine, tangent, and cotangent for angles between 90 and 180 degrees are negative, although they have the same absolute values as their respective angles from 0 to 90 degrees. Negative trigonometric values are preceded by a minus sign; thus, tan 123 ° 20 ′ = - 1.5204. Inverse Trigonometric Functions: If the value of the sides of a triangle are known but an angle is unknown, the trigonometric function inverse is used to work backwards to find the angle measure. Inverse trigonometric functions are known by either the prefix “arc” before the name, or by the superscript –1 after the name. The notation is not to be confused with the meaning of a negative exponent. Trigonometric functions and their inverses are shown below. The roles of x and y are re- versed in the inverse functions: x is the ratio given and y is the angle measure that is sought: Function Inverse Function y = sin x y = arcsin x, or y = sin –1 x y = cos x y = arccos x, or y = cos –1 x y = tan x y = arctan x, or y = tan –1 x The value of the angle that corresponds to the ratio is intended. Any angle can be found using the inverse trig function and relying on trig tables or a calculator. The examples give a few of the essential, familiar ratios for which the angles are readily known. Examples: Given sin x = 1, arcsin(1), or sin –1 (1) = 90°, or π /2. Given cos x = 0.866, arccos(0.866), or cos –1 (0.866) = 60°, or π /3. Given tan x = 1, arctan(1), or tan –1 (1) = 45°, or π /4. Trigonometry Tables.— Table 2a, Table 2b, and Table 2c, starting on page 108, contain the values of the sine, cosine, tangent, and cotangent functions of angles from 0 to 90 de- grees. Commonly referred to as “trig tables,” these values also are accessible on standard scientific calculators. Function values for all other angles can be obtained from the trig tables by applying the rules for signs of trigonometric functions and the useful relation ships among angles given in the following. Secant and cosecant functions can be found from sec A = 1/cos A and csc A = 1/sin A . The trig tables are divided by a double line. The body of each half table consists of four labeled columns of data between the columns that contain the angles. The angles left of the data increase moving down the table, and angles right of the data increase moving up the table. Column labels above the data identify the trig functions for angles listed in the left column of each half table. Columns labels below the data are for angles listed in the right column of each half table. To find the value of a function for a particular angle, first the angle is located in the table, then the appropriate function label across the top or bottom row of the table is located. At the intersection of the angle row and label column is the function value. Angles on opposite sides of each table are complementary angles (i.e.,
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