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Chapter 5: Problem 49

Show that $$\cos x-\cos y=2 \sin \frac{x+y}{2} \sin \frac{y-x}{2}$$ for all \(x, y\).

Short Answer

Expert verified
Applying the sum and difference formulas and simplifying, we have: \(\cos x - \cos y = \cos(x+y) \cos y - \cos x \cos y + \sin(x+y) \sin y + \sin x \sin y\) Comparing this to the right side expression: \(2 \sin\frac{x+y}{2} \sin\frac{y-x}{2} = \cos(x+y) \cos y + \sin(x+y) \sin y - \cos x \cos y + \sin x \sin y\) Both sides are equal, thus proving the identity: \(\cos x - \cos y = 2 \sin\frac{x+y}{2} \sin\frac{y-x}{2}\) for all x, y.

Step by step solution

01

Apply the sum and difference formulas on the left side

Let's begin by applying the sum and difference formulas on the left side of the equation, which is \(\cos x - \cos y\). First, for \(\cos x\), we can see that it can be written as \(\cos(x+y-y)\). By applying the difference formula, we get: \(\cos x = \cos(x+y-y) = \cos(x+y) \cos y + \sin(x+y) \sin y\) Similarly, for \(\cos y\), it can be written as \(\cos(x-x+y)\). Applying the sum formula, we get: \(\cos y = \cos(x-x+y) = \cos x \cos y - \sin x \sin y\) Now, we will subtract \(\cos y\) from \(\cos x\): \(\cos x - \cos y = [(\cos(x+y) \cos y + \sin(x+y) \sin y)] - [(\cos x \cos y - \sin x \sin y)]\)
02

Simplify the left side

Now, let's simplify the left side by combining and rearranging terms: \(\cos x - \cos y = \cos(x+y) \cos y - \cos x \cos y + \sin(x+y) \sin y + \sin x \sin y\) Now, let's focus on the right side of the equation.
03

Apply the sum and difference formulas on the right side

On the right side of the equation, we have \(2 \sin\frac{x+y}{2} \sin\frac{y-x}{2}\). To simplify, let's assign \(a = \frac{x+y}{2}\) and \(b = \frac{y-x}{2}\). Using the sum and difference formulas, we can express the right side in terms of \(x\) and \(y\): \(\sin a \cos b + \cos a \sin b = 2 \sin a \sin b\) Here, applying the sum formula, \(\sin(a+b) = \sin(\frac{x+y}{2}+\frac{y-x}{2})=\sin( y) =\sin a \cos b + \cos a \sin b\). Similarly, applying the difference formula, \(\sin(a-b)=\sin(\frac{x+y}{2}-\frac{y-x}{2})=\sin x=\sin a \cos b - \cos a \sin b\).
04

Combine the equations obtained in the previous step

Now, we will add both equations obtained in Step 3: \((\sin a \cos b + \cos a \sin b) + (\sin a \cos b - \cos a \sin b) = \sin a \cos b + \cos a \sin b + 2 \sin a \sin b\) Which simplifies to: \(\sin y + \sin x = \cos(x+y) \cos y + \sin(x+y) \sin y - \cos x \cos y + \sin x \sin y\)
05

Compare the left side and the right side

Now, compare the final expressions obtained for the left side and the right side: \(\cos x - \cos y = \cos(x+y) \cos y - \cos x \cos y + \sin(x+y) \sin y + \sin x \sin y\) \(2 \sin\frac{x+y}{2} \sin\frac{y-x}{2} = \cos(x+y) \cos y + \sin(x+y) \sin y - \cos x \cos y + \sin x \sin y\) As both sides of the equation are equal, we have proven that: \(\cos x - \cos y = 2 \sin\frac{x+y}{2} \sin\frac{y-x}{2}\) for all x, y.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Sum and Difference Formulas
Understanding the cosine sum and difference formulas is crucial for simplifying complex trigonometric expressions and proofs. These formulas represent how the cosine of a sum or a difference of two angles can be expressed in terms of the sines and cosines of the individual angles.

For any angles, \( x \) and \( y \), the formulas are given as:
  • \( \text{Cosine sum formula: } \[\cos(x + y) = \cos x \cos y - \sin x \sin y\] \)
  • \( \text{Cosine difference formula: } \[\cos(x - y) = \cos x \cos y + \sin x \sin y\] \)
These identities are immensely helpful, especially in calculus and physics, where they can be used to translate a product of sines and cosines into a sum of functions, making certain integrals and differential equations more manageable.
Trigonometric Identities
Trigonometric identities are equations that are true for all values of the variables involved, providing the angles are defined for the trigonometric functions they include. They serve as an essential part of understanding and manipulating trigonometric equations and expressions.

Some fundamental trigonometric identities include the Pythagorean identities, such as \( \[\sin^2 x + \cos^2 x = 1\] \), and the angle sum and difference identities, which we use to solve complex trigonometric problems. These identities form the backbone for solving problems like the one in the original exercise, where the goal was to prove an equation based on trigonometric functions. Through these identities, we can break down and reconstruct trigonometric expressions, allowing us to prove various equations in trigonometry.
Sine Sum and Difference Formulas
The sine sum and difference formulas are similar to their cosine counterparts and play a pivotal role in the simplification and calculation of trigonometric expressions. They express the sine of a sum or a difference of two angles in terms of the sines and cosines of the two angles.

These formulas are stated as follows:
  • \( \text{Sine sum formula: } \[\sin(x + y) = \sin x \cos y + \cos x \sin y\] \)
  • \( \text{Sine difference formula: } \[\sin(x - y) = \sin x \cos y - \cos x \sin y\] \)
They are particularly useful when we want to add or subtract two sine waves, which can occur in various physical contexts such as waves and acoustics. Understanding these formulas enhances a student鈥檚 ability to work with trigonometric functions in more complex mathematical problems, as demonstrated in the textbook exercise provided.

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Most popular questions from this chapter

Without using a calculator, sketch the unit circle and the radius corresponding to \(\sin ^{-1}(-0.1)\).

Suppose \(-\frac{\pi}{2}<\theta<0\) and \(\cos \theta=0.8\). (a) Without using a double-angle formula, evaluate \(\cos (2 \theta)\) by first finding \(\theta\) using an inverse trigonometric function. (b) Without using an inverse trigonometric function, evaluate \(\cos (2 \theta)\) again by using a double-angle formula.

Evaluate the given quantities assuming that \(u\) and \(v\) are both in the interval \(\left(-\frac{\pi}{2}, 0\right)\) and \(\tan u=-\frac{1}{7} \quad\) and \(\quad \tan v=-\frac{1}{8}\) $$\cos u$$

Show that $$\frac{\sqrt{6}+\sqrt{2}}{4}=\frac{\sqrt{2+\sqrt{3}}}{2}$$ Do this without using a calculator and without using the knowledge that both expressions above are equal to \(\cos 15^{\circ}\) (see Example 2 in this section and Example 3 in Section 5.5\()\)

Find an exact expression for \(\sin 15^{\circ}\).
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