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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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Most popular questions from this chapter

Challenge Problem Let the Dog Roam A dog is attached to a 9-foot rope fastened to the outside corner of a fenced-in garden that measures 6 feet by 10 feet. Assuming that the dog cannot enter the garden, compute the exact area that the dog can wander. Write the exact area in square feet.

What is the domain of the sine function?

Convert each angle to a decimal in degrees. Round your answer to two decimal places.\(40^{\circ} 10^{\prime} 25^{\prime \prime}\)

Find the exact value of \(\tan 40^{\circ}+\tan 140^{\circ} .\)

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