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Chapter 5: Problem 1

Fill in the blank. \( \sin\left(u - v\right) \) =________

Short Answer

Expert verified
\( \sin(u - v) = \sin(u)\cos(v) - \cos(u)\sin(v) \)

Step by step solution

01

Recall The Trigonometric Identity For Sine Difference

The formula representing the sine of the difference of two angles \(u\) and \(v\) is given by: \( \sin(u - v) = \sin(u)\cos(v) - \cos(u)\sin(v) \)
02

Substitute The Identity Into The Equation

Substituting the trigonometric identity into the given equation, we get: \( \sin(u - v) = \sin(u)\cos(v) - \cos(u)\sin(v) \). This completes the exercise as this formula is our solution.

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Key Concepts

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

Sine Difference Identity
The sine difference identity is an essential trigonometric identity that allows us to find the sine of the difference between two angles. This identity is especially helpful when dealing with complex angle calculations.
The formula is given by:
  • \( \sin(u - v) = \sin(u)\cos(v) - \cos(u)\sin(v) \)
This identity represents a precise way of breaking down the sine of a difference into simpler components involving sine and cosine.
The logic behind this formula comes from the geometry of the unit circle, helping to convert a single trigonometric function into others that may be easier to evaluate or understand in practical problems.
Angle Subtraction Formulas
Angle subtraction formulas are mathematical expressions used to calculate the trigonometric functions of the difference between two angles. They extend fundamental concepts in trigonometry by using known angle values to find new values.
For sine, the subtraction formula is:
  • \( \sin(u - v) = \sin(u)\cos(v) - \cos(u)\sin(v) \)
Similar formulas exist for cosine and tangent. These formulas make trigonometric calculations more accessible, especially when dealing with unfamiliar or complex angles:
  • Cosine: \( \cos(u - v) = \cos(u)\cos(v) + \sin(u)\sin(v) \)
  • Tangent: \( \tan(u - v) = \frac{\tan(u) - \tan(v)}{1 + \tan(u)\tan(v)} \)
These are powerful tools in solving problems involving angular displacements, oscillations, and wave equations in physics and engineering.
Trigonometric Formulas
Trigonometric formulas are mathematical equations involving trigonometric functions like sine, cosine, tangent, and their inverses. They are instrumental in connecting different aspects of triangles and circles.
  • Basic Formulas: These include \( \sin^2(u) + \cos^2(u) = 1 \), which helps establish relationships between sine and cosine for any angle \( u \).
  • Reciprocal Relations: Such as \( \csc(u) = \frac{1}{\sin(u)} \), \( \sec(u) = \frac{1}{\cos(u)} \), and \( \cot(u) = \frac{1}{\tan(u)} \).
  • Angle Addition and Subtraction: These allow for the calculation of functions like \( \sin(u \pm v) \), \( \cos(u \pm v) \), which are crucial for solving complex trigonometric equations.
Understanding these formulas helps solve problems in diverse fields from architecture to astronomy, and they form the basis of deeper exploration into calculus and analytical geometry.

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Most popular questions from this chapter

Exercises 43-52, use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. \( \cos^4 2x \)

In Exercises 81-90, use the product-to-sum formulas to write the product as a sum or difference. \( \sin (x + y) \cos(x - y) \)

In Exercises 77-80, find all solutions of the equation in the interval \( [0, 2\pi) \). Use a graphing utility to graph the equation and verify the solutions. \( \sin \dfrac{x}{2} + \cos x = 0 \)

In Exercises 103 - 106, find all solutions of the equation in the interval \( \left[0,2\pi\right) \). Use a graphing utility to graph the equation and verify the solutions. \( \cos 2x - \cos 6x = 0 \)

In Exercises 29-36, use a double-angle formula to rewrite the expression. \( 4 - 8 \sin^2 x \)
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