Expand Using Sum/Difference Formulas tan(105 degrees ) Step 1. First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, can be split into . Step 2. Use the sum formula for tangent to simplify the expression. The formula states that . Step 3.Sum and Difference Formulas. Mario's Math Tutoring. 300. views. 02:32. Using the Sum and Difference Identities for Sine, Cosine and Tangent, Ex 3. patrickJMT. 340. views. 02:40. Sum and Difference Identities for Sine and Cosine, More Examples #2. patrickJMT. 380. views. Showing 1 of 12 videos. Load more videos.21 May 2010 ... Visit http://mathispower4u.wordpress.com/ for a categorized and searchable list of all videos.You don't have to be a mathematician to follow this simple value statement formula. Trusted by business builders worldwide, the HubSpot Blogs are your number-one source for educati...Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. Sum and Difference Formulas. Mario's Math Tutoring. 300. views. 02:32. Using the Sum and Difference Identities for Sine, Cosine and Tangent, Ex 3. patrickJMT. 340. views. 02:40. Sum and Difference Identities for Sine and Cosine, More Examples #2. patrickJMT. 380. views. Showing 1 of 12 videos. Load more videos.Dec 12, 2022 · B: Evaluate sum and difference formulas given trig ratios of angles. Exercise \(\PageIndex{B}\) 41. Given that \(\sin a=\dfrac{4}{5}\) and \(\cos b=\dfrac{1}{3 ... The process for factoring the sum and difference of cubes is very similar to that for the difference of squares. We first identify \(a\) and \(b\) and then substitute into the appropriate formula. The separate formulas for sum and difference of cubes allow us to always choose \(a\) and \(b\) to be positive. Example \(\PageIndex{6 ...Feb 8, 2013 · Sine, cosine, or tangent of two angles that are added or subtracted. Sum and Difference Identities. With your knowledge of special angles like the sine and cosine of30 ∘ and45 ∘, you can find the sine and cosine of15 ∘, the difference of45 ∘ and30 ∘, and75 ∘, the sum of45 ∘ and30 ∘ . Using what you know about the unit circle and ... Course: Class 11 math (India) > Unit 3. Lesson 5: Trigonometric identities: Sum and difference. Trig angle addition identities. Using the cosine angle addition identity. Using the cosine double-angle identity. Trigonometric functions of sum and difference of angles. Proof of the sine angle addition identity. Proof of the cosine angle addition ...👉 Learn how to evaluate the cosine of an angle in radians using the sum/difference formulas. To do this, we first express the given angle as a sum or a diff...By utilizing the complex plane, I can easily derive the double angle formulas in my head, and quickly develop the sum and difference formulas on paper. You need only understand that multiplying by a complex number amounts to scaling and rotation in the plane.The formula to friendship. Steven Strogatz in The New York Times answers the question of why your Facebook friends always seem to have more friends than you. In a colossal study of...Differentiation of Sums and Differences; Examples. Example 1; Example 2; Example 3; Example 4; Review; Review (Answers) Vocabulary; Additional Resources; Based on …The sum and difference formulas for sine and cosine can also be used for inverse trigonometric functions. See Example \(\PageIndex{4}\). The sum formula for tangent states that the tangent of the sum of two angles equals the sum of the tangents of the angles divided by \(1\) minus the product of the tangents of the angles. The …Ex-Lax Maximum Relief Formula (Oral) received an overall rating of 4 out of 10 stars from 2 reviews. See what others have said about Ex-Lax Maximum Relief Formula (Oral), including...We will begin with the sum and difference formulas for cosine, so that we can find the cosine of a given angle if we can break it up into the sum or difference of two of the special angles. Sum formula for cosine. cos(α + β) = cosαcosβ − sinαsinβ. cos ( α + β) = cos α cos β − sin α sin β. Difference formula for cosine. Trigonometry. Expand Using Sum/Difference Formulas tan (285) tan(285) First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, 285 can be split into 225 + 60. tan(225 + 60) Use the sum formula for tangent to simplify the expression. The formula states that tan(A + B) = tan(A) + tan(B) 1 ...We can use the sum and difference formulas to identify the sum or difference of angles when the ratio of sine, cosine, or tangent is provided for each of the …The sum-to-product formulas allow us to express sums of sine or cosine as products. These formulas can be derived from the product-to-sum identities. For example, with a few substitutions, we can derive the sum-to-product identity for sine. Let u+v 2 = α u + v 2 = α and u−v 2 = β u − v 2 = β.The Formula: Tangent. The sum and difference angle formula for the tangent function is: Notice the formula has a plus-minus sign and a minus-plus sign. When we are dealing with a sum of two angles, the numerator will contain an addition sign but the denominator will contain a subtraction sign. The opposite is true when we are dealing …Companies can sell common stock shares to raise funds, but it’s important to first know how much you stand to gain from such a sale. Calculating the value of common stock can be do...Companies can sell common stock shares to raise funds, but it’s important to first know how much you stand to gain from such a sale. Calculating the value of common stock can be do...CK12-FoundationThe difference formulas for sine and cosine can be derived easily from the sum formulas, using the identities for negative angles. Note that the difference formulas are identical to the corresponding sum formulas, except for the signs. 2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple. 4 min read • january 29, 2023. Welcome back to AP Calculus with Fiveable! We are now diving into some of the most valuable fundamental concepts in calculus that allow us to find derivatives for complex polynomial functions more easily.How to use the Sum and Difference Formulas to find the exact value of the sine, cosine or tangent of an angle.0:12 Formulas for sine, cosine and tangent sum ...Microsoft Excel is a powerful business tool as it gives you the ability to calculate complex numbers and create intricate formulas. For instance, you can calculate the sum of multi...The sum and difference formulas for sine can be derived in the same manner as those for cosine, and they resemble the cosine formulas. A General Note: Sum and Difference Formulas for Sine These formulas can be used to calculate the sines of sums and differences of angles.Exercise Set 6.1: Sum and Difference Formulas MATH 1330 Precalculus 555 Answer the following. 43. Use a sum or difference formula to prove that TTsin sin . 44. ¨¸Use a sum or difference formula to prove that Answer the following. (Hint: It may help to first draw TTcos cos . 45. Use a sum or difference formula to prove that TTtan tan . 46.Learn how to use the sum and difference formulas for sine, cosine, and tangent functions with examples and derivations. The formulas state that the sum and difference of any two angles are related by the cosine and sine of the sum and difference angles. Learn the formulas for functions of the sum or difference of two angles. BrownMath.com → Trig w/o Tears → 7 / Sum and Difference ... Summary: Continuing with trig identities, this page looks at the sum and difference formulas, namely sin(A ± B), cos(A ± B), and tan(A ± B). Remember one, and all the rest flow from it. There’s also a ...Sum and Difference of Angles Identities. Angle sum identities and angle difference identities can be used to find the function values of any angles however, the most feasible use of sum of angles trig identities is to identify the exact values of an angle that can be mathematically expressed as a sum or difference using the familiar values …The sum and difference formulas for sine and cosine can also be used for inverse trigonometric functions. See Example \(\PageIndex{4}\). The sum formula for tangent states that the tangent of the sum of two angles equals the sum of the tangents of the angles divided by \(1\) minus the product of the tangents of the angles. The …Deciding between breastfeeding or bottle-feeding is a personal decision many new parents face when they are about to bring new life into the world. Deciding between breastfeeding o...29 May 2023 ... Let's see how we can learn it 1.In sin, we have sin cos. In cos, we have cos cos, sin sin In tan, we have sum above, and product below 2.W e can use the sum and difference formulas to solve trigonometric equations. For this concept, we will only find solutions in the interval 0 ≤ x < 2 π. Solve the function using the sum and difference formulas: cos (x − π) = √ 2 2. Use the formula to simplify the left-hand side and then solve for x.Verifying Trigonometric Identities. Sum and Difference Formulas. Double-Angle, Power-Reducing, and Half-Angle Formulas. Product-to-Sum and Sum-to-Product Formulas. Trigonometric Equations. 7. Additional Topics in Trigonometry. The Law of Sines.This trigonometry video tutorial explains how to use the sum and difference identities / formulas to evaluate sine, cosine, and tangent functions that have angles that …Companies can sell common stock shares to raise funds, but it’s important to first know how much you stand to gain from such a sale. Calculating the value of common stock can be do...Notice that we don’t really need a formula to factor a difference or sum of cubes! a) 8 125x3 + = +(2 ) 5x 3 3 = + − +(2 5)(4 10 25)x x x2 Practice Factor the following: ... Microsoft PowerPoint - Factoring Sums and Differences of Cubes - For …B: Evaluate sum and difference formulas given trig ratios of angles. Exercise \(\PageIndex{B}\) 41. Given that \(\sin a=\dfrac{4}{5}\) and \(\cos b=\dfrac{1}{3 ...This paper makes three original contributions: (1) Explicit closed-form parametric formulas for the boundary of the Minkowski sum and difference of two arbitrarily oriented solid ellipsoids in n-dimensional Euclidean space are presented; (2) Based on this, new closed-form lower and upper bounds for the volume contained in …Step 2. Now let us prove the formula of the cosine of the sum of angles. We represent the sum of angles (α + β) as difference (α– (–β)). Then, we use the formula of the cosine of the difference of angles obtained at step 1: The addition Formulas of the cosine angles are proved. The cosine of a sum of two angles.Trigonometry Sum and Difference Formulas. The sum identity is a formula that expresses the sine of the sum of two angles as a function of the sine and cosine of the individual angles. The most commonly used sum identity is the sine addition formula, which states that: sin (α + β) = sin α cos β + cos α sin β. where α and β are any two ...The difference formulas for sine and cosine can be derived easily from the sum formulas, using the identities for negative angles. Note that the difference formulas are identical to the corresponding sum formulas, except for the signs. Now the sum formula for the sine of two angles can be found: sin(α + β) = 12 13 × 4 5 + ( − 5 13) × 3 5 or 48 65 − 15 65 sin(α + β) = 33 65. 3. Find the exact value of sin15 ∘. Recall that there are multiple angles that add or subtract to equal any angle. Choose whichever formula that you feel more comfortable with.You can find the distance between two points by using the distance formula, an application of the Pythagorean theorem. Advertisement You're sitting in math class trying to survive ...All of the identities that relate the trig ratios of different angles are derived from the sum and difference formulas. Let’s see why we need these formulas. Is it true that cos(α +β) …The difference formulas for sine and cosine can be derived easily from the sum formulas, using the identities for negative angles. Note that the difference formulas are identical …Notice that we don’t really need a formula to factor a difference or sum of cubes! a) 8 125x3 + = +(2 ) 5x 3 3 = + − +(2 5)(4 10 25)x x x2 Practice Factor the following: ... Microsoft PowerPoint - Factoring Sums and Differences of Cubes - For …To divide by the sum of cells A1 through A10 by 2 in Excel, use the formula: =SUM(A1:A10)/2. It is also possible to enter numbers directly into the formula. When entering a formula...This trigonometry video tutorial explains how to find the exact of trigonometric expressions with angles in radians and degrees using the sum and difference ...Is there a scientific formula for funny? Read about the science and secrets of humor at HowStuffWorks. Advertisement Considering how long people have pondered why humor exists -- a...Here we'll get some extra practice putting the sum and difference formulas to good use. If you haven't gone through them yet, you might want to review the sections on the Sum and Difference Formulas for sine, cosine, and tangent. Solve using the Sum Formula Verify the identity \(\dfrac{\cos (x-y)}{\sin x \sin y}=\cot x \cot y+1\)Sum and difference formula with decimals - Trig. Ask Question Asked 6 years, 3 months ago. Modified 6 years, 3 months ago. Viewed 716 times 0 $\begingroup$ Can someone please walk me through the steps on how I can get an answer for a decimal problem involving sum and difference formulas? Thank you very much ...Jan 5, 2024 · Using the Sum and Difference Formulas, we can find these exact trig values. Sum and Difference Formulas. sin (a ± b) = sin a cos b ± cos a sin b cos (a ± b) = cos a cos b ∓ sin a sin b tan (a ± b) = tan a ± tan b 1 ∓ tan a tan b. Find the exact values using the Sum and Difference Formulas: sin 75 ∘ SUM AND DIFFERENCE FORMULAS FOR COSINE. These formulas can be used to calculate the cosine of sums and differences of angles. cos(α + β) = cosαcosβ − sinαsinβ. cos(α − β) = cosαcosβ + sinαsinβ. How to: Given two angles, find the cosine of the difference between the angles. Write the difference formula for cosine.The distinction between the two formulas is in the location of that one "minus" sign: For the difference of cubes, the "minus" sign goes in the linear factor, a − b; for the sum of cubes, the "minus" sign goes in the quadratic factor, a2 − ab + b2. Some people use the mnemonic " SOAP " to help keep track of the signs; the letters stand for ...Is there a scientific formula for funny? Read about the science and secrets of humor at HowStuffWorks. Advertisement Considering how long people have pondered why humor exists -- a...Using the Sum and Difference Identities for Sine, Cosine and Tangent. Example 3: Simplify 1 - 16sin 2 x cos 2 x. Show Video Lesson. Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. The female boss of the W Series says it's time to try something new to get women on the championship grid. Formula 1 is one of the few sports in which both sexes are free to compet...Learn how to use sum and difference formulas for cosine, sine, tangent, and cofunctions. See examples of how to apply the formulas to calculate the exact values of angles, distances, and functions. Explore the history …Sum and Difference Identities. First look at the derivation of the cosine difference identity: cos(α − β) = cosαcosβ + sinαsinβ. Start by drawing two arbitrary angles α and β. In the image above α is the angle in red and β is the angle in …The Pythagorean Theorem along with the sum and difference formulas can be used to find multiple sums and differences of angles. See . The cofunction identities apply to complementary angles and pairs of reciprocal functions. See . Sum and difference formulas are useful in verifying identities. See and .Sum and Difference of Trigonometric Functions Formulas. The sum and difference formulas help us evaluate the value of trigonometric functions at angles that can be expressed as the sum or difference of specific angles. In this guide, you will learn more about the sum and difference formulas.The sum and difference identities of angles are trigonometric identities, which can be used to find the values of trigonometric functions of any angle. However, the most practical use of these identities is to find the exact values of an angle that can be written as a sum or difference of familiar values for sine, cosine, and tangent of the angles 30°, 45°, 60°, …Trigonometric Sum, Difference, Product Identities & Equations: UVU Math Lab . Many of the following identities can be derived from the Sum of Angles Identities using a few simple tricks. ... Heron’s Formula: 𝐾𝐾= 𝑠𝑠(𝑠𝑠−𝑎𝑎)(𝑠𝑠−𝑎𝑎) ...Step 2. Now let us prove the formula of the cosine of the sum of angles. We represent the sum of angles (α + β) as difference (α– (–β)). Then, we use the formula of the cosine of the difference of angles obtained at step 1: The addition Formulas of the cosine angles are proved. The cosine of a sum of two angles.The straight-line depreciation formula is to divide the depreciable cost of the asset by the asset’s useful life. Accounting | How To Download our FREE Guide Your Privacy is import...B: Evaluate sum and difference formulas given trig ratios of angles. Exercise \(\PageIndex{B}\) 41. Given that \(\sin a=\dfrac{4}{5}\) and \(\cos b=\dfrac{1}{3 ...How to use the SUM function in Excel to add individual values, cell references, ranges, or a mix of all three. ... You can add and subtract times in a few different ways. For example, to get the difference between 8:00 AM - 12:00 PM for payroll purposes you would use: = ...Excel formula to find sum of difference. I have excel sheet with 2 columns: baseline, current. the baseline if base prices and the current is real price. Now, I need the difference if HIGHER current prices.. in our example, the first and last line have bigger current price, so i need the formula to return 6 (=15-10+2-1).Tangent Sum and Difference Formulas. In this lesson, we want to find a formula that will make computing the tangent of a sum of arguments or a difference of arguments easier. As first, it may seem that you should just add (or subtract) the arguments and take the tangent of the result. However, it's not quite that easy.The sum and difference formulas for tangent are: tan(α + β) = tanα + tanβ 1 − tanαtanβ. tan(α − β) = tanα − tanβ 1 + tanαtanβ. How to: Given two angles, find the tangent of the sum of the angles. Write the sum formula for tangent. Substitute the given angles into the formula. Simplify.It would be very helpful for students if they strive to derive the double-angle, half-angle, etc., formulas from the sum/difference formulas. Proving other ...Companies can sell common stock shares to raise funds, but it’s important to first know how much you stand to gain from such a sale. Calculating the value of common stock can be do...We can use the sum and difference formulas to identify the sum or difference of angles when the ratio of sine, cosine, or tangent is provided for each of the …The sum or difference of two even functions was an even function. The sum or difference of two odd functions was an odd function. The sum or difference of an even and odd function was neither even nor odd. These statements hold true in general as long as f and g are non-zero functions. Try to prove these results algebraically. — cos(x cos(x ...About this unit. In this unit, you'll explore the power and beauty of trigonometric equations and identities, which allow you to express and relate different aspects of triangles, circles, and waves. You'll learn how to use trigonometric functions, their inverses, and various identities to solve and check equations and inequalities, and to ... Use a sum or difference formula to find the exact value of cos 165°. 165° is not a special angle on the unit circle, but 165° = 135° + 30°, which are both special angles on the unit circle. So the sum formula for cosine can be used where = 135° and = 30°. $$\cos 165° = \cos\left (135° + 30°\right)$$. $$ = \cos 135° \cos 30° - \sin ...Using the Cosine Difference Formula. 1. Find an equivalent form of cos(π 2 − θ) using the cosine difference formula. cos(π 2 − θ) = cosπ 2cosθ + sinπ 2sinθ cos(π 2 − θ) = 0 × cosθ + 1 × sinθ, substitute cosπ 2 = 0 and sinπ 2 = 1 cos(π 2 − θ) = sinθ. We know that is a true identity because of our understanding of the ...Derivation of Sum and Difference of Two Angles. The sum and difference of two angles can be derived from the figure shown below. sin β = EF¯ ¯¯¯¯¯¯¯ 1; EF¯ ¯¯¯¯¯¯¯ = sin β sin β = E F ¯ 1; E F ¯ = sin β. tan(α + β) = sin α cos α + sin β cos β 1 − sin α cos α sin β cos β tan ( α + β) = sin α cos α + sin β ...Here we'll get some extra practice putting the sum and difference formulas to good use. If you haven't gone through them yet, you might want to review the sections on the Sum and Difference Formulas for sine, cosine, and tangent. Solve using the Sum Formula. Verify the identity cos (x − y) sin x sin y = cot x cot y + 1Sum and Difference Formulas: Introduction: In this lesson, formulas involving the sum and difference of two angles will be defined and applied to the fundamental trig functions. The Lesson: For two angles a and b, we have the following relationships: Sum formulas: sin(a + b) = sin(a)cos(b) + cos(a)sin(b) ...We will begin with the sum and difference formulas for cosine, so that we can find the cosine of a given angle if we can break it up into the sum or difference of two of the special angles. Sum formula for cosine. cos(α + β) = cosαcosβ − sinαsinβ. cos ( α + β) = cos α cos β − sin α sin β. Difference formula for cosine. 18 Mar 2013 ... Dive into the fascinating world of trigonometry as you explore the proof of the sum and difference formulas. Enhance your understanding of ...Sum and Difference Identities. First look at the derivation of the cosine difference identity: cos(α − β) = cosαcosβ + sinαsinβ. Start by drawing two arbitrary angles α and β. In the image above α is the angle in red and β is the angle in …However, it's not quite that easy. To find the sum formula for tangent: tan(a + b) = sin(a + b) cos(a + b) Using tanθ = sinθ cosθ = sinacosb + sinbcosa cosacosb − sinasinb Substituting the sum formulas for sine and cosine = sinacosb + sinbcosa cosacosb cosacosb − sinasinb cosacosb Divide both the numerator and the denominator …
So everywhere we saw a y here, we can replace it with a -y. So this is going to be equal to tangent of x plus the tangent of -y, all of that over 1 minus the tangent of x times the tangent of -y. Well, we know the tangent of -y is the same thing as the negative tangent of y. And we know that over here as well. . I believe in santa
Use the sum of angles formulas for sine and cosine to derive a formula for each expression. Then use graphs to verify your formula. [latex]\displaystyle \cos (\theta + 90°)[/latex] [latex]\displaystyle \sin (\theta + 90°)[/latex] 74. Use the sum of angles formulas for sine and cosine to derive a formula for each expression.Sum and Difference Formulas. Double-Angle, Power-Reducing, and Half-Angle Formulas ... Using the Sum and Difference Identities for Sine, Cosine and Tangent, Ex 1 ... However, it's not quite that easy. To find the sum formula for tangent: tan(a + b) = sin(a + b) cos(a + b) Using tanθ = sinθ cosθ = sinacosb + sinbcosa cosacosb − sinasinb …Jan 5, 2024 · Using the Sum and Difference Formulas, we can find these exact trig values. Sum and Difference Formulas. sin (a ± b) = sin a cos b ± cos a sin b cos (a ± b) = cos a cos b ∓ sin a sin b tan (a ± b) = tan a ± tan b 1 ∓ tan a tan b. Find the exact values using the Sum and Difference Formulas: sin 75 ∘ To divide by the sum of cells A1 through A10 by 2 in Excel, use the formula: =SUM(A1:A10)/2. It is also possible to enter numbers directly into the formula. When entering a formula...The sum and difference formulas for sine can be derived in the same manner as those for cosine, and they resemble the cosine formulas. A General Note: Sum and Difference Formulas for Sine These formulas can be used to calculate the sines of sums and differences of angles. Abu al-Wafa’s proof of the sum formulas for the sine and cosine functions. A little cranky for students, but interesting. Home. News Feed. Resources. Profile. People. Classroom. App Downloads. Sum and difference formulas. Author: Joseph Manthey. New Resources. Thin Slice: Exploring Right Triangles; Thin Slice: Special Triangle Theorems (1B)Toggle Angle sum and difference identities subsection. 3.1 Sines and cosines of sums of infinitely many angles. 3.2 Tangents and cotangents of sums. ... Ptolemy's theorem is …Jun 15, 2022 · Differentiation of Sums and Differences. Here are the differentiation rules for the sum and difference of two functions: d dx[f(x) + g(x)] = d dx[f(x)] + d dxddx[g(x) and. d dx[f(x) − g(x)] = d dx[f(x)] − d dx[g(x)] In simpler notation (f ± g)′ = f′ ± g′. Using the limit properties of previous chapters should allow you to figure out ... Notice that we don’t really need a formula to factor a difference or sum of cubes! a) 8 125x3 + = +(2 ) 5x 3 3 = + − +(2 5)(4 10 25)x x x2 Practice Factor the following: ... Microsoft PowerPoint - Factoring Sums and Differences of Cubes - For …By utilizing the complex plane, I can easily derive the double angle formulas in my head, and quickly develop the sum and difference formulas on paper. You need only understand that multiplying by a complex number amounts to scaling and rotation in the plane.Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. sin (π/2 – x) = cos x. Now we have the idea about the expansion of sum and difference of angles of cos. Now let us try to use it for finding the values of sum and difference of angles of sin. sin (x + y) can be written as cos [π/2 – (x + y)] which is equal to cos [ (π/2 – x) – y] Now, using identity (2) we can write,CAGR and the related growth rate formula are important concepts for investors and business owners. In this article, we'll discuss all you need to know about CAGR. Let's get started...Check your high school student's understanding of TRIG FORMULAS with this completely editable QUIZ. Two different versions are included in the folder as well as the keys. What's included:TWO different versions of the quiz Quiz KEYS Editable Word file for the quizTopics Covered: Sum and Difference FormulasDouble and Half-Angle FormulasIn …Toggle Angle sum and difference identities subsection. 3.1 Sines and cosines of sums of infinitely many angles. 3.2 Tangents and cotangents of sums. ... Ptolemy's theorem is important in the history of trigonometric identities, as it is how results equivalent to the sum and difference formulas for sine and cosine were first proved.Nov 19, 2021 · But it’s traditional to present all four formulas. Sum to Product. There are also formulas that combine a sum or difference into a product. Heon Joon Choi, a physics student from Cornell, has kindly told me of an application: “superposing two waves and trying to figure out the nodes is much easier if they are multiplied, rather than added.” .