Geometry with Trigonometry
240 Trigonometric functions in calculus Ch. 12 Definition. Given any angleα∈A∗(F), in 9.4.1 we defined^12 α.Nowforany integern≥ ...
Sec. 12.2 Circular functions 241 Without loss of generality we suppose that|α|◦≤|β|◦,sothatbytheabove, |^12 α|◦≤|^12 β|◦.As|^12 ...
242 Trigonometric functions in calculus Ch. 12 The functions c and s have the following properties:- (i)For 0 ≤x≤ 360 , 0 ≤y≤ 36 ...
Sec. 12.2 Circular functions 243 The sequences un,vnhave the following properties:- (i)(un)is non-decreasing. (ii)(vn)is non-inc ...
244 Trigonometric functions in calculus Ch. 12 COMMENT. This equation forμis referred to asDarboux’s functional equa- tion. Ther ...
Sec. 12.3 Derivatives of cosine and sine functions 245 must exist. But the formulaμ(^1 n)=μ(n^1 )shows thatμ(^1 n)→ 0 (n→∞)and s ...
246 Trigonometric functions in calculus Ch. 12 Proof. Suppose that 0<x<360, andh>0 is so small thatx+h< 360 ,x−h> ...
Sec. 12.4 Parametric equations for a circle 247 12.4.2Lengthofanarcofacircle ...................... For P∈C(O;k)letαbe an angle ...
248 Trigonometric functions in calculus Ch. 12 12.5 Extensionofdomainsofcosineandsine ............... 12.5.1 ................... ...
Sec. 12.5 Extension of domains of cosine and sine 249 so for some complex constantc, E(x+y)=cE(x),for allx∈R. On puttingx=0 we s ...
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List of axioms AXIOM A 1 .Each line is a proper non-empty subset ofΠ. For each set{A,B}of two distinct points inΠ, there is a un ...
252 List of axioms (iii) if∠BAC is a wedge-angle and the point D=A lies in the interior region IR(|BAC),then |∠BAD|◦+|∠DAC|◦=|∠ ...
Bibliography [1] T.M. Apostol.Calculus, volume I and II. Blaisdell, second edition, 1967. [2] J.K. Baumgart et al., editors.Hist ...
Index Z 3 Z 4 Z 1 Z 2 , 116 (O,Z 1 )+(O,Z 2 ), 188 (O,Z 1 ).(O,Z 2 ), 190 (Z 1 ,Z 2 )↑(Z 3 ,Z 4 ), 185 −αd, 175 (^0) F, 135 (^0) ...
Index 255 −→ OZ ⊥ , 207 c(x), 241 f(F), 127 i(α), 130 l≡Fax+by+c=0, 87 l‖m,54 l⊥m,44 nα, 138 rα;Z 0 , 150 s(x), 241 sα;Z 0 , 153 ...
256 Index diameter, 105 exterior point, 103 interior point, 103 major arc, 114 minor arc, 114 nine-point, 225 non-secant line, 1 ...
Index 257 equipollent point pairs, 185 erected perpendicular, 44 Euclidean geometry, 61 Euler line, 215 exterior point circle, 1 ...
258 Index circle, 105 norm vector, 190 null duo-angle, 169 null vector, 191 null-angle, 30 open half-plane, 26 opposite vertex a ...
Index 259 degree-measure, 46 reflex-angle, 30 region exterior, 30 interior, 29 right duo-angle, 169 right-angle, 44 rotation, 15 ...
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