In basic plane trigonometry, trigonometric ratios of Sine or Cosine of an angle can have any value between -1 and + 1 and these can not be more than 1 or less than -1. To define sine of an angle, it is ratio of perpendicular to hypotenuse and cosine of an angle is ratio of base to hypotenuse in a right angled Triangle. From these definitions also and since hypotenuse is always larger than any of other side, these ratios are natural to be less than one. A number of problems have been solved by all assuming sin (n.x) and cos (n.x) to be equal to any value between -1 and + 1 when angle n.x tends to infinite with n approaching infinity. Assumption of (sin x)^n or (cos x)^n as approaching zero when these ratios are not equal to one with n tending to Infinity has always been made. All these assumptions are based on fact that magnitude of trigonometric ratios of sine and cosine are always less than or equal to one.

Are such assumptions then incorrect in view of the statement that these ratios can be larger than one? It is unmathematical to have answer in “yes,” and “no” to same question. This, therefore, needs analysis to reach logical result about the magnitude of these ratios. For that, it will be examined what deters to equate these trigonometric ratios to quantities larger than one.

Let us assume sin x equal to a quantity larger than one and let it be square root 5. Cos x can be calculated from identity (1- sin^2 x)^1/2 and that makes cos x equal to (-4)^1/2 or 2i where i is (-1)^1/2 and is called imaginary number. A question then arises whether imaginary quantity is practical and realisable. Simple answer to this query is “yes” if it is assumed that imagination is the result of real quantity. If a real quantity can give birth to imaginary then reverse should also be true and imagination should also give birth to reality.

*Friedrich August Kekulé,* a German chemist saw in his dream that snakes are entangled eating tails of each other and that led him to proclaim valency of carbon four and the structure of benzene. Apart from this, *imaginary number raised to power imaginary number* ie i^i equals e^-pi/2 *Here base is imaginary and power is also imaginary but the result is real and astonishing as it relates to pi which is the worlds most important mathematical constant.

Imaginary quantity can, therefore, not be dubbed as insignificant as it is analogous to *operator 90 degree. *That is it rotates the real quantity by 90 degree, for example 2i means, a quantity of magnitude 2 rotated by 90 degree, in other words, it is a quantity of magnitude 2 that lies on y axis. Similarly, i.i or -1 means rotation by 180 degree or – x axis,

i.i.i or -i means rotation by 270 degree or -y axis and

i.i.i.i or 1 means rotation by 360 degree or x axis.

This analogy of imaginary number is utilised in electrical engineering where current leads or lags according to capacitive or inductive nature of load and is represented by C.i or -C.i where C is the current flowing in the load. It can then be summed up, imaginary quantity is as important, significant and realisable as real quantity.

Coming to cos x = 2i calculated from Pythagorean equality, it is quite apparent that angle x must be complex so as to make cos x as imaginary quantity. A complex quantity is that which has real and imaginary part. If ‘a’ and ‘b’ are real then a + i.b is a complex quantity where a can be zero also. *Let us assume angle *

*x = a+ i.b, then
*

cos (a+ i.b) = 2.i

or cos a. cos i.b – sin a. sin i.b

= cos a.cosh b – i sin a.sinh b,

since cos ia = cosh a and sin ia = i.sinh b.

Therefore,

cos a.cosh b – i sin a.sinh b = 2.i.

Equating real and imaginary parts,

cos a.cosh b= 0,

i sin a.sinh b = 2i

or sin a.sinh b = 2.

That means *a =(2k + 1). pi/2 or (2k- 1).pi/2.,Substituting this value,
*

1.sinh b = 2 or b = sinh inverse 2.

Sinh b can also be written as ½.(e^b – e^-b) = 2.

On putting e^b= y, we get

y – 1/y = 4 or y^2- 4y – 1= 0.

That gives

y = e^b = 2 + 5^1/2 or 2 – 5^1/2.

This can also be written as

y = 2Φ+ 1 or 3- 2Φ

where Φ is golden ratio equal to

½.(1+ 5^1/2).

From this, *b can be written
*

*= log (2 + 5^1/2) or log (2 – 5^1/2) or log (2Φ+ 1) or log (3- 2Φ)
*

Therefore *complex angle whose cosine is 2i and sine is square root 5 is
*

*= Pi/2 +i.log (2 + 5^1/2) or Pi/2 +i.log (2Φ+ 1).
*

On taking the principal value of a and positive value of b, we can also write

Sin [Pi/2 +i.log (2 + 5^1/2)] = 5^1/2,

Or sin [Pi/2 +i.log (2Φ+ 1)]= 5^1/2

Or cosh [log (2Φ+ 1)] = 2Φ-1,

cosh [log (2 + 5^1/2)] = 5^1/2,

Also sinh (log (2 + 5^1/2) = 2

or sinh [log (2Φ+ 1)] = 2.

From above analysis, we can say, cosine or sine of an angle can be larger than one provided the angle is complex containing imaginary part. Imaginary part is not an imagination but is realisable by way of cosh and sinh of imaginary angle as given below.

Cos ix = cosh x = ½.(e^x + e^-x),

sin ix = i.sinh x = ½.i.(e^x – e^-x)

and this leads to Pythagorean identities

*cos^2 x + sin^2 x = cosh^2 x – sinh^2 x = 1.
*

Next question that arises is how to draw a complex or imaginary angle. *In Plane Trigonometry, real angle can be drawn in x-y Plane and trigonometric ratios can be found out by drawing a right angled triangle in that plane. When an angle is drawn in a plane, it always has real value.That means, it can never be complex and resultantly cosine and sine of an angle as ratios of base to hypotenuse and perpendicular to hypotenuse can never be larger than one as hypotenuse in a Plane trigonometry is always larger than its sides. Therefore sine and cosine of an angle is always equal or less than one in plane trigonometry.
*

Coming to drawing of an imaginary angle, I submit, such a case must satisfy identity cosh^2 x – sinh^2 = 1 and in Plane trigonometry, imaginary angle would be such that the straight lines forming it, never join. When the lines do not join, it can never be defined as an angle. Therefore, in a Plane x-y where real angle can be drawn, imaginary angle can never be drawn.

*That necessitates there should be another plane an if that Plane is considered perpendicular to the plane that contains real angle, then imaginary part of the angle can be drawn in that perpendicular plane. Let us see how. It is well known to those who have basic knowledge of plane trigonometry that as the angle increases the length rotates in a circle. Or we can say that if length starts from point (0,0) and ends at a point (a cos x, a sin x), then with increase in angle x, the line rotates in a circle of radius a in counter clockwise directin. The locus of the point in Plane trigonometry with change in angle is circle. Why locus is a circle can be proved from the identity
*

cos^2 x + sin^2 = 1

On multiplying both sides by a^2, we get

a^2.cos^2 x + a^2. Sin^2 x = a^2,

and that is an equation of circle. It is reiterated that when angle is real in plane trigonometry, a point can be defined as a point on a circle.

It is explicit from the explanation already given that when angle changes to imaginary angle say i.x then cosine of imaginary angle takes the form cos i x and that is equal to cosh x and cosh x is equal to ½(e^x + e^-x). When angle is imaginary, sine of imaginary angle takes the form sin ix and that is equal to i sinh x and i sinh x is equal to 1/2.i(e^x -e^-x). Therefore, it leads to identity,

cosh^2 x – sinh^2 x = 1.

On multiplying both sides by a^2, we get

a^2.cosh^2 x –a^2. sinh^2 x = a^2.

Let a^2.cosh^2 x be x and sinh^2 x = y, then locus of point x, y represents

x^2 – y^2 = 1

and that is an equation of hyperbola. That is why cosh x and sinh x are called hyperbolic functions.

It follows from this, *change in imaginary angle drawn on perpendicular plane moves the point a.cosh x, a sinh x on hyperbola. We can, therefore, conclude that whenever there is trigonometric ratio and if that pertains to complex angle, real part of the angle will decide the length on the circumference of a circle in a plane and its imaginary part the length on the hyperbola in a plane perpendicular to the plane of real angle.
*

Based on this analysis let me answer the questions I raised at the start of this paper. *Sine and cosine of a real angle that lies in a plane is always either equal to or less than one. It is true in plane trigonometry what is taught in schools. Length of the line rotates in a circle as the real angle is changed if we keep magnitude of length constant.
*

When the angle is complex (that is it has imaginary part also), sine and cosine of complex angle can be larger than one and in that case, imaginary part of the angle lies in a plane perpendicular to the in which real angle lies and the point moves upon hyperbola drawn in perpendicular plane with the change in imaginary angle. Finally, sine or cosine of an angle is Plane trigonometry is equal or less than one but if it pertains to complex angle containing imaginary part, it can be greater than one.

End

References:

1. Title Image courtesy Kan8eDie at https://upload.wikimedia.org/wikipedia/commons/e/eb/Principle_branch_of_arg_on_Riemann_%28small%29.png

2.Kekulé Dream by Michael Verderese

Professor Heinz D. Roth at https://web.chemdoodle.com/kekules-dream/

3.* Eternal Bonds Of Togetherness Between Natural Number, Pi, Tangent And Logarithm Of A Quantity at https://narinderkw.wordpress.com/2017/08/12/eternal-bonds-of-togetherness-between-natural-number-pi-tangent-and-logarithm-of-a-quantity/

4.Image of complex angle courtesy Richard Hammack at A Geometric View of Complex Trigonometric Functions http://www.people.vcu.edu/~rhammack/reprints/cmj210-217.pdf

About Author:

Writer is an Electronics and Electrical Communication Engineering graduate and was earlier Scientist, then Instrument Maintenance Engineer, then Civil Servant in Indian Administrative Service (IAS). After retirement, he writes on subjects, Astronomy, Mathematics, Yoga, Humanity etc.