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 = 4 + 2 cos t, y = 3 + 5 sin t; t = π/2

Verified step by step guidance

1. Identify the given parametric equations: \(x = 4 + 2 \cos t\) and \(y = 3 + 5 \sin t\).
2. Substitute the given parameter value \(t = \frac{\pi}{2}\) into the expression for \(x\): calculate \(x = 4 + 2 \cos \left( \frac{\pi}{2} \right)\).
3. Substitute the same parameter value \(t = \frac{\pi}{2}\) into the expression for \(y\): calculate \(y = 3 + 5 \sin \left( \frac{\pi}{2} \right)\).
4. Evaluate the trigonometric functions \(\cos \left( \frac{\pi}{2} \right)\) and \(\sin \left( \frac{\pi}{2} \right)\) using known unit circle values.
5. Combine the results from the previous steps to find the coordinates \((x, y)\) of the point on the curve corresponding to \(t = \frac{\pi}{2}\).

Key Concepts

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 like circles or ellipses.

Evaluating Trigonometric Functions at Specific Angles

To find coordinates for a given parameter t, you substitute t into the trigonometric functions (cos t and sin t). Knowing exact values of sine and cosine at common angles like π/2 is essential for accurate calculation of points on the curve.

Coordinate Calculation from Parametric Form

Once the parameter value is substituted, calculate x and y by performing arithmetic operations on the trigonometric results. This yields the precise point (x, y) on the plane curve corresponding to the given t.