Question

cosa+ sina=√2 cosa.show that cosa -sina =√2 sina​

Answer 1

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1

Cours d’analyse –Chap. VII: Imaginary expressions1/153Chapter VIIOn imaginary expressions and their moduli._______§ I –General considerations on imaginary expressionsIn analysis, we call a symbolic expressionor symbolany combination of algebraic signs that do not mean anything by themselves or to which we attribute a value different from that which they naturally have. Likewise, we call symbolic equations all those that, taking the letters and the interpretations according to the generally established conventions, are inexact or do not make sense, but from which we can deduce exact results in modifying and altering them according to fixed rules either the equations themselves or the symbols which comprise them. The use of symbolic expressions or equations is often a means of simplifying calculations and of writing in a short form results that appear quite complicated. We have already seen this in the second paragraph of the third chapter where formula (9) gives a very simple symbolic value to the unknown xsatisfying the equations (4). Among the symbolic expressions or equations the consideration of which is of some importance in analysis, we should especially distinguish those which we call imaginary. We are going to show how we can put them to good use.We know that the sine and the cosine of the arc a + bare given as functions of the sines and cosines of the arcs aand bby the formulas (1)cosabcosacosbsinasinbsinabsinacosbsinbcosa/154But, without taking the pain to remember these formulas, we have a very simple means of recovering them at will. It suffices, in fact, to consider the following remark.Suppose that we multiply together the two symbolic expressionscosa1sina,cosb1sinb,

Cours d’analyse –Chap. VII: Imaginary expressions2by applying the known rules of algebraic multiplication and knowing that -1 is a real [?? Actual?] quantity the square of which is equal to -1. The resulting product is composed of two parts. The one real and the other has a factor -1. The real part gives the value of cos(a + b) and the coefficient of -1 gives the value of sin(a + b). To observe this remark, we write the formula(2)cosab1sinabcosa1sinacosb1sinbThe three expressions that make up the preceding equation, namelycos a+ -1 sin a,cos b+ -1 sin b,cos(a + b) + -1 sin(a + b),are three symbolic expressions that cannot be interpreted using the generally established conventions, and they do not represent anything real. For this reason, they are called imaginary expressions. The equation (2) itself, taken literally, inexact and it does not make sense. To get exact results, first we must expand its second part by algebraic multiplication, and this reduces the expression to(3)cosab1sinabcosacosbsinasinb1sinacosbsinbcosaSecondly, we must equate the real part of the first member (left hand side) of equation (3) with the real part of the righthand side, then the coefficient of -1 on the left hand side with the coefficient of -1 on the right. Thus we recover the equations (1) both of which we ought to consider as implicitly contained in the formula (2)In general, we call an imaginary expressionany symbolic expression of the form+ -1,where and denote real quantities. We say that two expressions + -1 and + -1are equalto each other when there is equality between corresponding parts, that is between the real parts and , and between the coefficients of -1, namely and . We indicate the equality between two imaginary expression in the same way that we it between two real quantities, by the symbol =, and this results in what we call an imaginary equation. Thissaid, any imaginary expression is just the symbolic representation of two equations between real quantities. For example, the symbolic equation + -1 = + -1