One More Path To Trigonometric Addition Formulas

There’s something truly beautiful about the internal consistencies of mathematics. Thanks to its nature we get to hike on such a variety of paths to reach the same destination.  Math becomes a mental traveler’s paradise.

Consider the trigonometric addition formulas:

sin(x + y) = sin(y) cos(x) + sin(x) cos(y)  and

cos (x + y) = cos (x) cos(y)  –  sin(x) sin(y) .

These lead to highly useful double angle formulas, which are needed to evaluate integrals. They are also used in Fourier analysis and in communication systems. How do we derive them? I’ll first explore a slightly more difficult route. Secondly we will briefly refer to a conventional geometrical approach and finally we’ll examine my own offbeat path (but I can’t guarantee it’s original) .

(1) A circuitous route using slightly higher mathematics

Knowing the rate of change of sine, cosine and natural base functions allows one to express sin(w), cos(w) and ew as power series. Replace the w with xi, where i is the square root of -1, the so-called imaginary number, and exi turns out to be the sum of the cosine series and i times the sine series, namely Euler’s relationship:

exi = cos (x) + i sin(x).

If we subsequently use the above to express ei(x+y)  we obtain

ei(x+y) = cos (x + y) + i sin(x + y).                        equation(1)

But using exponent laws, ei(x+y) = exi (eyi ) , and then applying Euler’s formula again:

ei(x+y) = exi [eyi ] = [cos(x) + i sin(x) ] [cos(y) + i sin(y) ] and expanding the right hand side:

ei(x+y) = cos(x) cos(y) + i sin(y) cos(x) +  i sin(x) cos(y)  – sin(x) sin(y).

If we regroup the above by pairing and separating the real components from the imaginary ones :

ei(x+y) = cos(x) cos(y)   – sin(x) sin(y) + i [sin(y) cos (x) + sin(x) cos (y)].        equation(2)

we can compare equations (1) and (2) and have to conclude that:

cos(x + y) = cos(x) cos(y)  –  sin(x) sin(y).

sin(x + y) = sin(y) cos(x) + sin(x) cos(y).

(2) A Geometric Approach

Elementary math can be used if we begin with the following diagram. The details of this particular derivation are found at the site indicated in the caption.

TrigAnglesWeisstein_1000
http://mathworld.wolfram.com/TrigonometricAdditionFormulas.html

(3) A simpler off beat path from working backwards

After deriving the Sine Law from simple trig ratios, I subsequently thought of using the same diagram to make substitutions into the sine addition formula. Surely enough, after some manipulations, I obtained the Sine Law, implying that the latter can be used to derive the other.

IMG_1169
A derivation of the sine addition formula

By using the same approach with the cosine addition formula, I obtained another unexpected starting point. Apologies for not typing it out. The convenience of simply photographing my handwritten versions proved to be too tempting!

IMG_1170
A derivation of the cosine addition formula
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Why Soft Sounds Are Pleasurable

At the other extreme of physical pain, are soft, tactile impressions that are just above the threshold of perception. Imagine a gentle wind finding its way through body hair or the lightest of drizzles falling on one’s bare shoulders. The acoustic equivalent of such pleasures is the distant sound of rustling leaves. At about 10 decibels it has less than 10 times the intensity of the faintest audible sound. And it has only one of ten trillionths of the power per area of a sound wave that could cause pain. It seems that we have evolved not only to react to extreme stimuli that endanger us but also to be rewarded when the environment toys with the lower limits of our senses.

Although the above numbers are mathematically accurate—-they are based on the formula which equates decibels with 10 times the logarithmic ratio of a sound’s intensity to our threshold intensity of 10-12 watts/meter2—they exaggerate what our eardrums actually experience.

pressurewave
from hyperphysics.phy-astr.gsu.edu  Check link for neat demo idea using methane in pipe to visualise pressure variations.

Before delving further, just what is sound? When a leaf moves back and forth, or some other body vibrates, the oscillation causes a periodic disturbance of the surrounding air molecules. The movement in one direction causes molecules of the air (or other transmitting medium) to bunch up, increasing pressure in one region. As the object reverts to its original position, a region of less crowding among molecules occurs as well. Meanwhile, the crowded region transmits its kinetic energy to other particles of the medium, repeating the pattern of high and low densities as the pressure wave propagates away from its source.

As a result, what’s more appropriate in comparing sound-strength is the use of pressure amplitude, which is proportional to the square root of a sound’s intensity. For example if our slightly rustling leaves come in at 10 dB and a pneumatic sidewalk-breaker next to us is painfully wreaking havoc at 120 dB, first we figure out the intensity ratio from the exponential version of the decibel-log formula:

I2/I1 = 10 0.1(dB2 – dB1) =10 0.1(120 – 10) = 1011.

Then by taking the square root  of 1011, we obtain the pressure ratio— about 316 000. That’s still a huge factor for how much louder the pneumatic drill is in comparison to the distant sound of gently moving leaves.

A little more number-crunching can make us appreciate how wonderfully sensitive our ears are, at least when they are working well. The exact pressure amplitude(ΔPm) of a rustling sound can be calculated from:

ΔPm = √(2Iρv) ,

where I is the intensity, and ρ and v are the density and velocity of the medium. For air at room temperature the latter numbers are 1.2 kg per meter cubed and 343 m/s, and for a 10 dB-sound, I = 10-11 W/m². Thus ΔPm is  9.07 X 10-5 pascals. Comparing it to the atmospheric pressure of 101.3 kPa, we can actually detect a difference of 1 part in a billion, and it’s enough to give us pleasure! When you consider that it takes a few parts per million of psychotropic substances to buzz the brain, something that many marvel at, to me our acute sensitivity to sound seems more remarkable, while no drugs are necessary.

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