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Chapter 6: Problem 2

Write each product as a sum or difference of sines and/or cosines. $$\cos (10 x) \sin (5 x)$$

Short Answer

Expert verified
\( \cos(10x) \sin(5x) = \frac{1}{2}[\sin(15x) - \sin(5x)] \)

Step by step solution

01

Identify the Formula

When faced with a product of sine and cosine, we use the product-to-sum identities. For \( \cos A \sin B \), the identity is: \[ \cos A \sin B = \frac{1}{2}[(\sin(A + B) - \sin(A - B))]. \] We will use this identity to express \( \cos(10x) \sin(5x) \) as a sum or difference.
02

Apply the Identity

Substitute \( A = 10x \) and \( B = 5x \) into the product-to-sum formula. This gives us: \[ \cos(10x) \sin(5x) = \frac{1}{2}[(\sin((10x) + (5x)) - \sin((10x) - (5x)))]. \] Simplifying these expressions inside the sine functions results in: \[ \cos(10x) \sin(5x) = \frac{1}{2}[(\sin(15x) - \sin(5x))]. \] This is the expression as a sum and difference of sines.

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Key Concepts

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

Trigonometric Identities
Trigonometric identities are equations that relate the trigonometric functions. These identities are essential tools in trigonometry, as they help simplify expressions and solve equations efficiently. You may hear about different types of identities such as Pythagorean, reciprocal, and co-function identities.
But in the context of our exercise, we highlight the product-to-sum identities. These identities convert products of sine and cosine into sums or differences, making complex expressions easier to handle.
  • The product-to-sum identity for the product of cosine and sine is: \( \cos A \sin B = \frac{1}{2} [\sin(A + B) - \sin(A - B)] \).
  • This form is useful for transforming the expression \( \cos(10x) \sin(5x) \) into a sum or difference, which may simplify further analysis or integration.
Mastery of these identities is vital in advancing through trigonometry and calculus.
Sine and Cosine Functions
The sine and cosine functions are fundamental building blocks in trigonometry. They describe oscillations and periodic phenomena, appearing in varied fields such as physics, engineering, and even music.
The sine function, \( \sin(x) \), represents the y-coordinate of a point on the unit circle, which makes it a periodic function with a cycle of \( 2\pi \). Its range spans from -1 to 1.
  • The cosine function, \( \cos(x) \), similarly reflects the x-coordinate on the unit circle. It shares the sine's periodicity, but leads with a phase shift.
  • Both functions are closely related, demonstrated by key identities like \( \sin^2(x) + \cos^2(x) = 1 \), expressing their complementary nature.
In the exercise, by expressing the product \( \cos(10x) \sin(5x) \) as a sum or difference, we explicitly decompose the intertwined behaviors of these two functions.
Angle Sum and Difference Formulas
The angle sum and difference formulas are valuable tools in trigonometry, used to express the sine or cosine of angle sums or differences as products of trigonometric functions of simpler angles.
For example, the sine identity for angle sum is \( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \). This guides us in reorganizing complex expressions into more manageable parts.
  • In our context, understanding these formulas helps simplify \( \sin(15x) - \sin(5x) \), coming from the product-to-sum identity used in the exercise.
  • These formulas are essential for deriving the product-to-sum identities mentioned earlier.
By grasping the angle sum and difference formulas well, students can navigate through expressions requiring reevaluation with more expertise, greatly facilitating efforts in solving complicated trigonometric problems.

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Most popular questions from this chapter

Solve the trigonometric equations exactly on the indicated interval, \(0 \leq x<2 \pi\). $$\cos (2 x)=\sin x$$

Solve the trigonometric equations exactly on the indicated interval, \(0 \leq x<2 \pi\). $$2 \sin x+\csc x=3$$

Although in general the statement \(\sin (A+B)=\sin A+\sin B\) is not true, it is true for some values. Determine some values of \(A\) and \(B\) that make this statement true.

Solve the trigonometric equations exactly on the indicated interval, \(0 \leq x<2 \pi\). $$\csc x-\cot x=\frac{\sqrt{3}}{3}$$

Write \(\sin A \sin B \sin C\) as a sum or difference of sines and cosines.
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