By D. C. Champeney
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Extra info for Fourier Transforms and Their Physical Applications
36 and Equation (1), we see that the central angle u measured in radians is just the length of the arc that the angle cuts from the unit circle. 017) radians. 1 shows the equivalence between degree and radian measures for some basic angles. 37). Angles measured counterclockwise from the positive x-axis are assigned positive measures; angles measured clockwise are assigned negative measures. 37 Angles in standard position in the xy-plane. Angles describing counterclockwise rotations can go arbitrarily far beyond 2p radians or 360°.
Dividing this identity in turn by cos2 u and sin2 u gives 1 + tan2 u = sec2 u 1 + cot2 u = csc2 u The following formulas hold for all angles A and B (Exercise 58). Addition Formulas cos (A + B) = cos A cos B - sin A sin B sin (A + B) = sin A cos B + cos A sin B (4) There are similar formulas for cos (A - B) and sin (A - B) (Exercises 35 and 36). All the trigonometric identities needed in this book derive from Equations (3) and (4). For example, substituting u for both A and B in the addition formulas gives Double-Angle Formulas cos 2u = cos2 u - sin2 u sin 2u = 2 sin u cos u (5) Additional formulas come from combining the equations cos2 u + sin2 u = 1, cos2 u - sin2 u = cos 2u.
37 Angles in standard position in the xy-plane. Angles describing counterclockwise rotations can go arbitrarily far beyond 2p radians or 360°. 38). y y y y 5p − 2 3p x x 9p 4 Angle Convention: Use Radians From now on, in this book it is assumed that all angles are measured in radians unless degrees or some other unit is stated explicitly. When we talk about the angle p>3, we mean p>3 radians (which is 60°), not p>3 degrees. We use radians because it simplifies many of the operations in calculus, and some results we will obtain involving the trigonometric functions are not true when angles are measured in degrees.