Ch. 3 - Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 3 - Derivatives

Problema 56a

Capitolo 3, Problema 56a

Assume that a particle’s position on the x-axis is given by

x = 3 cos t + 4 sin t,

where x is measured in feet and t is measured in seconds.

a. Find the particle’s position when t = 0, t = π/2, and t = π.

Guida verificata passo dopo passo

To find the particle's position at a specific time, substitute the given values of t into the position function x = 3 cos t + 4 sin t.

For t = 0, substitute t = 0 into the equation: x = 3 cos(0) + 4 sin(0). Recall that cos(0) = 1 and sin(0) = 0.

For t = π/2, substitute t = π/2 into the equation: x = 3 cos(π/2) + 4 sin(π/2). Recall that cos(π/2) = 0 and sin(π/2) = 1.

For t = π, substitute t = π into the equation: x = 3 cos(π) + 4 sin(π). Recall that cos(π) = -1 and sin(π) = 0.

Evaluate each expression to find the particle's position at t = 0, t = π/2, and t = π using the trigonometric values provided.

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Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in describing periodic phenomena. In this context, they represent the particle's oscillatory motion along the x-axis. Understanding how to evaluate these functions at specific angles, like 0, π/2, and π, is crucial for determining the particle's position at given times.

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Percorso guidato

6:04

Introduction to Trigonometric Functions

Evaluating Trigonometric Expressions

To find the particle's position at specific times, we need to evaluate the trigonometric expression x = 3 cos t + 4 sin t. This involves substituting the given values of t into the expression and calculating the result. Familiarity with the unit circle and the values of sine and cosine at key angles is essential for this process.

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Percorso guidato

6:04

Introduction to Trigonometric Functions

Position Function

The position function x = 3 cos t + 4 sin t describes the particle's location on the x-axis over time. It combines the effects of two harmonic motions, each with its amplitude and phase. Understanding how to interpret and manipulate this function allows us to predict the particle's position at any given time t.

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Percorso guidato

5:20

Relations and Functions

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