Parametric equations and a value for the parameter t are...

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 $t.$

$x=4+2 \cos t, y=3+5 \sin t ; t=\frac{\pi}{2}$

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Key Concepts

Parametric Equations

Parametric equations define a curve by expressing the x and y coordinates as functions of an independent parameter, usually denoted as t. This representation allows for describing curves that may not fit the standard y = f(x) form, and it is especially useful when dealing with complex or closed curves.

Parameter Evaluation

Parameter evaluation involves substituting a specific value for the parameter t into the given parametric equations to determine the corresponding point on the curve. This technique translates the abstract description encoded by the parameter into a concrete coordinate pair on the Cartesian plane.

Trigonometric Functions in Parametric Forms

Trigonometric functions such as cosine and sine are frequently used in parametric equations to represent curves with circular-like behavior, like circles and ellipses. Their periodic nature enables a natural depiction of repetitive or oscillatory motion, and they are fundamental in converting angular measures into Cartesian coordinates.

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