Date modified: Mon 2023-09-18 . 09:01 AM
Date created: Sat 2023-10-07 . 09:02 PM
# How to multiply numbers - Part 2 Trigonometric identities and prosthaphaeresis In the last part, you proved one of the most amazing fact about complex multiplication, that given $z=a+ib=r(\cos \alpha+i\sin \alpha)$, and $w=c+id=s(\cos \beta+i\sin \beta)$, we have their product $$ zw=(ac-bd)+i(ad+bc)=rs(\cos(\alpha+\beta)+i\sin(\alpha+\beta)) $$ where the magnitudes multiply together and the angles add together for the product $zw$. This is the **geometric interpretation of complex multiplication**. With this, we can easily establish trigonometric identities such as angle addition formulas, double angle formulas, etc! Let us see how this work! ![[---images/---assets/---icons/question-icon.svg]] Take the complex number $z=\cos \alpha+i\sin \alpha$, and $w=\cos \beta+i\sin \beta$. Calculate $zw$ in two ways: (1) Directly multiply out the components by distribution, and (2) interpret what the resulting magnitude and angles should be using the **geometric interpretation** of complex multiplication. Now match the real part with the real part, and the imaginary part with the imaginary part. What is $\cos(\alpha+\beta)$ and $\sin(\alpha+\beta)$ in terms of $\cos \alpha$, $\sin \alpha$, $\cos \beta$, $\sin \beta$? **Congratulations! You have just derived the angle addition formula!!** ///check/// Just to check, you should obtain: $\cos(\alpha+\beta)=\cos \alpha \cos \beta-\sin \alpha \sin \beta$ $\sin(\alpha+\beta)=\cos \alpha \sin \beta+\sin \alpha \cos \beta$ /// ![[---images/---assets/---icons/question-icon.svg]] Use your angle addition formula above, derive the **double angle formulas** for $\cos(2\alpha)$ and $\sin(2\alpha)$. ![[---images/---assets/---icons/question-icon.svg]] Also by changing $\beta$ to $-\beta$, one can get **angle difference formulas**! Recall cosine is an even function, and sine is an odd function[^parity]. ![[---images/---assets/---icons/question-icon.svg]] Can you derive the half angle formulas as well, what is $\cos\left( \frac{\alpha}{2} \right)$ and $\sin\left( \frac{\alpha}{2} \right)$? By the way, if you want the **tangent** versions of these, just remember $\tan \theta= \frac{\sin \theta}{\cos \theta}$. The angle addition formulas are a type of "sum-of-inputs to product-of-outputs" formula, that it takes inputs that are sums, and returns a product of outputs. We can also solve for the product-of-outputs to get sum-of-inputs as below: ![[---images/---assets/---icons/question-icon.svg]] Using the angle addition and difference formulas above, derive the following "product-sum" formulas: (1) $\cos(\alpha)\cos(\beta) =\frac{1}{2}(\cos(\alpha+\beta)+\cos(\alpha-\beta))$ (2) $\sin(\alpha)\sin(\beta)=\frac{1}{2}(\cos(\alpha-\beta)-\cos(\alpha+\beta))$ (3) $\sin(\alpha)\cos(\beta)=\frac{1}{2}(\sin(\alpha+\beta)+\sin(\alpha-\beta))$ Neat! ## Prosthaphaeresis. Now we are ready to get back to the story of multiplication of big numbers and how ancient mathematicians attempted to simplify their calculations, by using these "product-sum" formulas. The idea is that "adding" is easier than "multiplication", so we trade a multiplication problem with addition problems. Here is how it works. Suppose we have two numbers we want to multiply, $x$ and $y$, and let us say for the moment $x,y\in[-1,1]$, namely they are within the range of the cosine function. Then using a [[1 teaching/summer program 2023/week 3/cosine-table|cosine table]], find angles $\alpha$ and $\beta$ such that $\cos \alpha=x$ and $\cos \beta=y$. We can then compute $\alpha+\beta$ and $\alpha-\beta$ and use the "product-sum" formulas for cosine to calculate $xy=\cos(\alpha)\cos(\beta)$ ! Ok, here is a worked example. Let us say we want to multiply $x=0.73$ with $y=0.27$. (1) Using the [[1 teaching/summer program 2023/week 3/cosine-table|cosine table]], we see that $\alpha \approx 0.75$ and $\beta \approx 1.295$ (interpolate) such that $\cos \alpha \approx 0.73$ and $\cos \beta \approx 0.27$. (2) $\alpha+\beta=2.045$ and $\alpha-\beta=-0.545$ (3) Using the table again, we see $\cos(\alpha+\beta)\approx -0.455$ and $\cos(\alpha-\beta)\approx 0.8545$. Again, we interpolate in between values, and recall cosine is an even function. (4) Now we take the average of those to values, since we have the identity $$ \cos \alpha \cos \beta=\frac{1}{2} (\cos(\alpha +\beta) + \cos(\alpha-\beta)) $$ which we get $\frac{1}{2}(-0.455+0.8545) = \frac{0.3995}{2} \approx 0.19975$. This is the only step that we need to do division, but dividing by two tend to be easier. (5) And voila! We claim $xy=(0.73)(0.27)\approx 0.19975$. The actual value is of course $0.1971$. ![[---images/---assets/---icons/question-icon.svg]] How good was above approximation? And what would improve the accuracy of this method? ![[---images/---assets/---icons/question-icon.svg]] Try this method by multiplying $x= 0.14$ with $y=0.53$. Comment on your accuracy. ![[---images/---assets/---icons/question-icon.svg]] How would you modify this method to estimate the product of numbers that are bigger than one? Say $x=5123$ and $8828$? Hint: What you can do to move the decimals? ![[---images/---assets/---icons/question-icon.svg]] Use your modified method to estimate $5123\times 8828$. This procedure is called **prosthaphaeresis**, meaning "addition and subtraction", signifying the adding and subtracting the angles to compute this product. Ancient navigators and astronomers had trigonometric tables at their ready, so many used this method to compute repetitive products! ![[---images/---assets/---icons/question-icon.svg]] Suppose one measures something, and one gets a number with a lot of digits, say $58173712923$, versus another one just reports $58000000000$. Can you comment the difference of these two? What would these specific digits signify? This is actually an important concept in science and engineering called **significant digits**. ![[---images/---assets/---icons/question-icon.svg]] Comment on this process of prosthaphaeresis. What made multiplication easier? Or did it? What are some disadvantages? Later people sought another method to find products using sums, leading to the discovery of **logarithms**. Further questions: Can you adapt prosthaphaeresis to do **division** as well? Perhaps using another trigonometric table of values? === [^parity]: An even function is one such that $f(-x)=f(x)$, and an odd function is one such that $f(-x)=-f(x)$.