In Exercises 1–8, parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t. x = (60 cos 30°)t, y = 5 + (60 sin 30°)t − 16t²; t = 2

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Identify the parametric equations given: \(x = (60 \cos 30^\circ) t\) and \(y = 5 + (60 \sin 30^\circ) t - 16 t^2\).

Substitute the given value of the parameter \(t = 2\) into the equation for \(x\): calculate \(x = (60 \cos 30^\circ) \times 2\).

Substitute the same value \(t = 2\) into the equation for \(y\): calculate \(y = 5 + (60 \sin 30^\circ) \times 2 - 16 \times (2)^2\).

Recall the exact trigonometric values: \(\cos 30^\circ = \frac{\sqrt{3}}{2}\) and \(\sin 30^\circ = \frac{1}{2}\), and use these to simplify the expressions for \(x\) and \(y\).

After substituting and simplifying, write the coordinates of the point as \((x, y)\) corresponding to \(t = 2\).

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주요 개념

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Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves and motions.

Evaluating Trigonometric Functions at Specific Angles

Trigonometric functions like sine and cosine have known values at special angles such as 30°, 45°, and 60°. For example, cos 30° = √3/2 and sin 30° = 1/2. These values simplify calculations when substituting into parametric equations.

Substitution and Calculation of Coordinates

To find the point on the curve for a given parameter t, substitute the value of t into the parametric equations for x and y. This involves arithmetic operations and applying trigonometric values to compute the exact coordinates of the point.