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.

검증된 단계별 안내

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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주요 개념

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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.

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.

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.

Assume that a particle’s position on the x-axis is given byx = 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.