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Chapter 7: Problem 8

True or False The graph of the sine function has infinitely many \(x\) -intercepts.

Short Answer

Expert verified
True. The sine function does have infinitely many x-intercepts.

Step by step solution

01

Identify Definition

The first step is to understand what an x-intercept is. An x-intercept occurs where the function's graph crosses the x-axis. This happens when the output of the function is zero.
02

Analyze the Sine Function

The sine function, written as \(\text{sin}(x)\), has values that oscillate between -1 and 1. The sine function crosses the x-axis where \(\text{sin}(x) = 0\).
03

Find Intercepts of Sine Function

The sine function is zero at points \(\text{sin}(x) = 0\) when \(x = n\pi \) where \(n\) is any integer. This is because \(\text{sin}(0) = 0\), \(\text{sin}( \pi ) = 0\), \(\text{sin}(2 \pi ) = 0\), and so on.
04

Determine if Infinite Intercepts Exist

Since there are infinitely many integers, there will be infinitely many values of \(x\) for which \( \text{sin}(x) = 0\). Therefore, the graph of the sine function will have infinitely many x-intercepts.

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Key Concepts

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

x-intercepts
An x-intercept occurs where a function's graph crosses the x-axis. This happens when the output of the function, often noted as y, is zero. For example, if you have a function y=f(x), the x-intercepts are points (x,0) where f(x)=0. Identifying x-intercepts is crucial because they reveal where the function hits 'ground level鈥�. This concept is not only pivotal in algebraic functions but is equally essential in understanding trigonometric functions like the sine function.
sine function
The sine function is a basic trigonometric function, written as sin(x). It can be visualized as a wave that oscillates between -1 and 1. The key characteristics of this function include its periodic nature and specific values at certain points. For instance, sin(0) = 0, sin(蟺/2) = 1, and sin(蟺) = 0. The function repeats these values every 2蟺 intervals. This repetitive pattern helps in understanding where exactly the sine function crosses the x-axis, essential for finding x-intercepts.
trigonometric functions
Trigonometric functions, including sine, cosine, and tangent, are functions of an angle and are critical in studying triangles and oscillations. These functions have unique properties:- Periodicity: They repeat their values in regular intervals. For the sine function, its period is 2蟺.- Range: Sine and cosine functions oscillate between -1 and 1.- Zero points: Points where these functions cross the x-axis. In trigonometry, these properties facilitate solving various equations and modeling real-world phenomena like sound waves and circular motion.
infinite solutions
When we analyze the sine function, we find that it crosses the x-axis infinitely many times. The sine function equals zero at integer multiples of 蟺. Mathematically, this is expressed as sin(x)=0 when x=n蟺, where n is an integer. Because there are infinite integers, the number of x-values for which sin(x)=0 is also infinite. This characteristic demonstrates that the graph of the sine function has infinitely many x-intercepts, offering an excellent example of a function with infinite solutions in trigonometry.

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