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Trigonometrical Formulae

A collection of formulae covering, Addition and subtraction of Sin cos and tan; Product formulae ; the solution of equations and the half angle formulae and the Inverser Ratioi
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The Addition Formulae

13108/img_trig_a.jpg
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If OP and OQ are unit radii which make a angles with the x axis of B and A respectively. Then the coordinates of P are (Cos B ; sin B) and for Q (cos A ; sin A). By inspection the angle POQ is of magnitude A - B.

Using the Pythagora theorem,

Applying the cosine formula to triangle POQ:-

Equating equations (2) and (3)

This equation applies for all values of A and B

Writing (90^0\;+\;A) for A

If B is replaced by - B and making use of the fact that cos B = cos(-B) and that - sin B = sin(-B).Then:-

Putting A = B

The above equation can be expressed in two different forms:-

Equation (8) can be treated the same way in which case:-

par Addition Formulae for the Tangent.

Divide the Numerator and the denominator by \;cos\,A\;cos\,B

If B is replaced in the above equation by - B

From equation (16) it can be seen that :-

It is worth noting that :-

This is a particular case of the more general formlua

Where s_n stands for all the possible products of tan A ,tan B etc taken n at a time.

It follows from equation (21) that since the tan\,180^0\,=\;0 and if A, B, C are the angles of a triangle then:-

Useful Formulae

And

The Product Formulae.

By adding the two above equations we get:-

And by subtraction:-

In these two new equations we can substitute (A + B) = X and (A - B) = Y from which :-

Proceeding in a similar way we get:-

and

The Half Angle Formulae

By writing A = x/2 in formulae from the last sections :=

From equation (12)

And from (9) (10) (11)

and from equation (18)

These formulae allow us to express the sine; cosine; and tangent of an angle in terms of the tangent of the half angle. It is therefore possible to write

from which

Equation (36) can be re-written as :-

And from equation (37)

These three equations (40); (43) ; (46) are useful in the solution of a certain type of trigonometrical equation. They also have other important applications.

Example 1

If tan \theta\;=\;\frac{4}{3}\;and\;if\;0^0\;<\;\theta\;<\;360^0 find without tables the possible values of \,tan\,\frac{1}{2}\;\theta\;and\;of\;sin\,\frac{1}{2}\;\theta

Solving the quadratic:-

to find sin\,\frac{1}{2}\,\theta

The Auxiliary Angle

The equation a\;cos\,\theta\;+\;b\;sin\,\theta\;=\;c in which a; b; c are known numerical quantities . A method of solution is to divide throughout by \sqrt{(a^2\;+\;b^2)}

13108/img_trig_b.jpg
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If we introduce an angle \lambda whose tangent is b/a it can be seen that we can read off values for bot the sine and cosine. Hence the equation can be re-written as:-

The equation has now been reduced to one of the standard forms whose solution is known. Hence a value for \theta\;-\;\lambda can be found and as the value of \lambda is known \theta ca be calculated. For real solutions it is necessary for the value of c to be less than \sqrt{(a^2\;+\;b^2)}

A second method of solution is to use the half angle formulae ( Equations (43) and (46)

This quadratic gives two values for t from which general value of \theta can be found.

The Inverse Notation

If sin\theta = x where x is a given quantity numerically less than unity, wwe know that \theta can be any one of a whole series of angles. Thus if sin\,\theta\;=\;\frac{1}{2}\;,\;then,\theta\;=\;n\pi \;+\;(-\.1)^n(\frac{\pi }{6})\;and\;\theta can have a number of values. The inverse notation \theta\;=\;sin^{-1}\.x is used to denote the angle whose sine is x and the numerically smallest angle satisfying the relationship x\;=\;sin\,\theta is chosen as the principle value. Here and in what follows we shall deal only with principle values and the statement \theta \;=\;sin^{-1}\,x to mean that \theta is the angle that lies between -\frac{\pi }{2}\;and\;\frac{\pi }{2} radians whose sine is x. The statement \mathbf{\theta\;=\;sin^{-1}\,x} means that \theta is the inverse sine of x. On the continent this is sometimes written as \mathbf{\theta \;=\;arc\;sin\,x}

The graph of \theta\;=\;sin^{-1}\,x is, on thus that part of the graph x\;=\;sin\,\theta\;given\;by\;-\;\frac{\pi}{2}\; <\;\theta\;<\;\frac{\pi}{2} with the x-axis horizontal and the \theta axis vertical.As shown:-

13108/img_trig_c.jpg
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In a similar way \theta\;=\;cos^{-1}\,x will be taken to denote the smallest angle whose cosine takes the same value for negative as for positive angles and we require a notation which gives an unique value of \theta when x is given, we conventionally take \theta as the angle lying between 0 and \pi radians whose cosine is x.

For example

The graph of \theta\;=\;cos^{-1}\;x \;is \;derived \;from\; that \;of \;x\;=\;cos\,\theta.

13108/img_trig_d.jpg
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The inverse tangent is similarly defined but as, unlike the sine and cosine, the tangent can take all values, x is quite unrestricted in value. \theta\;=\;tan^{-1}\;x is taken to meantan^{-1}(1)\;=\;\frac{\pi }{4}\;\;\;and\;\;\;tan^{-1}\;(-\,1)\;=\;-\frac{\pi}{4}n that \theta lies between \frac{-\,\pi}{2}\;\;\;and\;\;\;\frac{\pi}{2} radians.

tan^{-1}(1)\;=\;\frac{\pi }{4}\;\;\;and\;\;\;tan^{-1}\;(-\,1)\;=\;-\frac{\pi}{4}

13108/img_trig_e.jpg
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It follows from these definitions that:-

These relationships will be found useful in some situations.

NOTE care must be taken avoid confusion between the inverse sine, cosine etc and the reciprocal of sin x, cos x etc. The latter should always be written as :-