Read e-book online Applied mathematics for radio and communication engineers PDF

By Carl E. Smith

Arithmetic, advisor, How-to, Communications, Engineering

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Adjacent side is A and the opposite side is C. I Since the ratios hold the same for a given angle regardless of the length of the three sides, we can always form this angle by using a protractor to determine the correct angle between two intersecting straight lines. Now, Opposite drop a perpendicular from a point Side on one of the lines to the other C=6 line to complete the right tri­ 90° angle. The three sides of the Adjacent Side triangle can then be measured A=8 and the ratios worked out from Ratio :l: = adjacent side = ~ = 0,8 these measurements.

This gives us a right triangle with vector P as the hypotenuse and having sides of length P", and P y. But this vector makes an angle a with the x-axis, so by trigonometry we have cos or Similarly, P", . sm or a = = a = Pi! = adj hyp = IPI cos opp hyp = IPI sin P", R", = P", (5) Py IPI a (6) sin {3 = opp = Qy hyp IQI Qy = IQI sin {3 (8) + Q", = IPI cos a + IQI cos {3 (9) Substituting Eqs. (6) and (8) in Eq. (2) gives R y = P y + Qy = IPI sin a + IQI sin {3 (10) Now, by substituting Eqs. (9) and (10) in (3), we get the resultant magnitude of the vector R in terms of the magnitude of vectors P, Q and trigonometric functions of the angles a and {3, that is, = v(IPI cos a + IQI cos (3)2 + (IPI sin a + IQI sin (3)2 (11) This equation is perfectly general for finding the magnitude of the vector R, which is the sum of any two vectors, P and Q, making angles a and {3 respectively with the positive x-axis.

12) is R. 485 Note: This is negative because it is in the second quadrant instead of the first. From trigonometric tables, 'Y = 82°56' This, however, is the angle vector R makes with the negative x-axis. If we subtract this from 180° we get the angle" vector R makes with the positive It will be noted that this equation is like Eq. (11) with the exception that If more vectors were to be added they ICI cos", and ICI sin", are added. x-axis, thus 'Y = 97°04' too would be included in each term under the radical, like C /"'.

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Applied mathematics for radio and communication engineers by Carl E. Smith


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