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Chapter 5: Problem 46

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

Short Answer

Expert verified
Using the cosine addition and subtraction formulas, we can write down the following equations: \[\cos (u+v) = \cos u \cos v - \sin u \sin v\] \[\cos (u-v) = \cos u \cos v + \sin u \sin v\] Subtracting the first equation from the second equation, we get: \[\cos (u-v) - \cos (u+v) = 2\sin u \sin v\] Finally, isolating the desired expression, we obtain: \[\frac{\cos (u-v) - \cos (u+v)}{2} = \sin u \sin v\] Thus, the given identity is true for all \(u, v\).

Step by step solution

01

Write down the cosine addition and subtraction formulas

The cosine addition formula is: \[\cos (a+b) = \cos a \cos b - \sin a \sin b\] The cosine subtraction formula is: \[\cos (a-b) = \cos a \cos b + \sin a \sin b\]
02

Replace a with u and b with v

By substituting \(a = u\) and \(b = v\), we get: \[\cos (u+v) = \cos u \cos v - \sin u \sin v\] \[\cos (u-v) = \cos u \cos v + \sin u \sin v\]
03

Subtract the first equation from the second equation

Subtract the first equation from the second equation: \[\cos (u-v) - \cos (u+v) = 2\sin u \sin v\]
04

Isolate the desired expression

Now, we need to isolate the desired expression \(\sin u \sin v\). We can do this by dividing both sides of the equation by 2: \[\frac{\cos (u-v) - \cos (u+v)}{2} = \sin u \sin v\] The proof is complete. We have shown that the given identity is true for all \(u, v\).

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Key Concepts

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

Sine Function
The sine function is one of the basic trigonometric functions, and it describes the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. If we consider a unit circle (a circle with radius 1 unit), the sine of an angle \( \theta \) is the y-coordinate of the point where the terminal side of the angle intersects the unit circle.

In terms of a formula, this relationship is expressed as \( \sin(\theta) = \frac{opposite}{hypotenuse} \). The sine function is periodic, with a period of \( 2\pi \) radians (or 360 degrees), and it varies between -1 and 1. This oscillatory function is crucial in modeling waves and oscillations in physics, and it is also widely used in engineering and mathematics.
Cosine Addition Formula
The cosine addition formula is an important identity in trigonometry which makes it possible to find the cosine of the sum of two angles. It is expressed as:

\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]

Applications:

  • Simplifying complex trigonometric expressions.
  • Calculating precise values of trigonometric functions for certain angles.
  • Solving trigonometric equations.
The formula shows that the cosine of a sum is not simply the sum of cosines; rather, it involves a mix of sine and cosine products and illustrates how these trigonometric functions are interrelated.
Cosine Subtraction Formula
Similar to the addition formula, the cosine subtraction formula plays a key role in trigonometry and is defined as:

\[ \cos(a - b) = \cos a \cos b + \sin a \sin b \]
By comparing it with the addition formula, we note that the only difference is the sign between the cosine and sine products. This change of sign reflects the nature of subtraction vs. addition of angles:

  • It provides the means to connect the cosine of a difference of angles to the product of sines and cosines of individual angles.
  • This identity is used in signal processing and in the derivation of other trigonometric formulas.
The symmetry presented in the subtraction and addition formulas is especially helpful in solving trigonometric equations and proving other identities.
Mathematical Proof
A mathematical proof is a logical argument that establishes the validity of a statement beyond any doubt using previously established facts, definitions, and logical operations. In the context of trigonometry, proving an identity, such as the one showing the relationship between the sine function and cosine functions of additions and subtractions of angles, requires a sequence of justified steps that lead to the desired conclusion.

Proving trigonometric identities often involves:
  • Utilizing known identities (like the cosine addition and subtraction formulas in our exercise).
  • Algebraic manipulation to simplify expressions.
  • Logical reasoning to deduce one fact from another.
Proofs not only validate the truth of a mathematical proposition but also encourage a deep understanding of the concept and its connection to other elements within mathematics.

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Most popular questions from this chapter

Find exact expressions for the indicated quantities. The following information will be useful: $$ \begin{array}{l} \cos 22.5^{\circ}=\frac{\sqrt{2+\sqrt{2}}}{2} \text { and } \sin 22.5^{\circ}=\frac{\sqrt{2-\sqrt{2}}}{2} \\ \cos 18^{\circ}=\sqrt{\frac{\sqrt{5}+5}{8}} \text { and } \sin 18^{\circ}=\frac{\sqrt{5}-1}{4} \end{array} $$ [The value for \(\sin 22.5^{\circ}\) used here was derived in Example 4 in Section \(5.5 ;\) the other values were derived in Exercise 64 and Problems 102 and 103 in Section \(5.5 .]\) $$\sin 82.5^{\circ}$$

Suppose \(-\frac{\pi}{2}<\theta<0\) and \(\cos \theta=0.3\). (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 indicated expressions assuming that $$ \begin{array}{ll} \cos x=\frac{1}{3} & \text { and } \sin y=\frac{1}{4} \\ \sin u=\frac{2}{3} & \text { and } \cos v=\frac{1}{5} \end{array} $$ Assume also that \(x\) and \(u\) are in the interval \(\left(0, \frac{\pi}{2}\right),\) that \(y\) is in the interval \(\left(\frac{\pi}{2}, \pi\right),\) and that \(v\) is in the interval \(\left(-\frac{\pi}{2}, 0\right) .\) $$\sin (x+y)$$

Show that $$\tan \frac{x}{4}=\frac{\sqrt{2-2 \cos x}-\sin x}{1-\cos x}$$ for all \(x\) in the interval \((0,2 \pi)\). [Hint: Start with a half-angle formula for tangent to express \(\tan \frac{x}{4}\) in terms of \(\sin \frac{x}{2}\) and \(\cos \frac{x}{2}\). Then use half-angle formulas for cosine and sine, along with algebraic manipulations.]

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