12 Trigonometric functions in calculus
COMMENT. In 3.7.1 and 9.3.1 we extended degree-measure to reflex angles, but
the only use we have made of this hitherto has been for the purposes of notation in
Chapter 9. In this chapter we relate it to the length of an arc of a circle, defining
πin the process. In doing this we make a start on calculus for the cosine and sine
functions, introduce radian measure of angles and can go on to derive Maclaurin
series. We also derive the area of a disk. In doing this we assume from the calculus
definition of the length of a rectifiable curve, the Riemann integral and a formula for
area in terms of a line integral; this is done so as to save labour, although it would be
possible to handle arc-length of a circle and the area of a disk more elementarily.
12.1 Repeatedbisectionofanangle ....................
12.1.1 .....................................
Addition⊕is associative onA∗(F).
Proof. On using 9.3.2 instead of 9.3.3, as in the proof of 9.3.3(iii) we see that
cos[(α⊕β)⊕γ]=cos[α⊕(β⊕γ)],
sin[(α⊕β)⊕γ]=sin[α⊕(β⊕γ)].Thus(α⊕β)⊕γandα⊕(β⊕γ)are the angles inA∗(F)with the same cosine
and the same sine, so they are the same angle except perhaps whenP 3 =Qand either
angle could be 0For 360F. Thus(α⊕β)⊕γ=α⊕(β⊕γ), except perhaps when
either side is a null or full angle. But when each ofα,β,γis 0F, then each of(α⊕
β)⊕γ,α⊕(β⊕γ)is 0Fandsotheyareequal.Ifanyofα,β,γis non-null, and
either of(α⊕β)⊕γ,α⊕(β⊕γ)is 0For 360Fthen both of them must be 360F,
and again they are equal.
Geometry with Trigonometry
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