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Chapter 6: Problem 22

What is the period of the function \(\sin (-5 x) ?\)

Short Answer

Expert verified
The period of the function \(\sin (-5x)\) is \(\frac{2\pi}{5}\).

Step by step solution

01

Identify the coefficient of the variable

The coefficient of the variable \(x\) in the given function \(\sin (-5x)\) is \(-5\).
02

Apply the period formula

We can use the formula \(T = \frac{2\pi}{|B|}\) to find the period of the sine function. Here, we have \(B = -5\). Replace \(B\) with \(-5\) in the formula: $$T = \frac{2\pi}{|-5|}.$$
03

Calculate the period

Now, evaluate the expression for \(T\): $$T = \frac{2\pi}{5}.$$ Thus, the period of the function \(\sin (-5x)\) is \(\frac{2\pi}{5}\).

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Key Concepts

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

Understanding the Sine Function
The sine function, denoted as \( \sin(x) \), is fundamental in trigonometry. It is one of the primary trigonometric functions along with cosine and tangent. The sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the triangle's hypotenuse. Its graph is a smooth, wave-like curve, repeating every \(2\pi\) radians, which is the function's period. This properties make the sine function periodic and continuous.
  • The sine function oscillates between -1 and 1.
  • A complete cycle of the sine wave from 0 to \(2\pi\) radians encompasses one period.
  • The graph of \( \sin(x) \) crosses the x-axis at multiples of \(\pi\), \(0, \pi, 2\pi, \ldots\)
Recognizing these features of the sine graph is crucial in understanding how trigonometric functions behave generally.
What is Trigonometry?
Trigonometry is the study of the relationships involving lengths and angles of triangles. It evolved from geometry and is mainly focused on right triangles. This branch of mathematics is essential for solving problems in engineering, physics, and other applied sciences. Trigonometry allows us to compute unknown distances or angles in various practical problems.
  • Trigonometry gives us functions like sine, cosine, and tangent.
  • These functions tell us the ratios of sides in a right-angled triangle for a given angle.
  • Beyond triangles, trigonometry helps describe circular and periodic phenomena.
Understanding trigonometry opens the door to advanced mathematical concepts, including calculus and complex numbers.
Transforming Functions: Sine Function Period
Function transformation involves changing the appearance or position of a function's graph. For the sine function, one of the key transformations is altering its period, which changes how often it completes a full cycle.
  • The standard period of sine \(\sin(x)\) is \(2\pi\).
  • When you have \(\sin(Bx)\), it changes the period to \(\frac{2\pi}{|B|}\).
  • In \(\sin(-5x)\), \(B = -5\), leading to a period of \(\frac{2\pi}{5}\).
This period change affects the frequency of the wave. Function transformation through period alteration helps model real-world periodic phenomena, such as sound waves or tides.

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