Question

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

Short answer

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

Step-by-step solution

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.

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 \(\text{sin}(x) = 0\).

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.

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.

Key concepts

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’. 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(π/2) = 1, and sin(π) = 0. The function repeats these values every 2π 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 2π.- 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 π. Mathematically, this is expressed as sin(x)=0 when x=nπ, 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.