Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. The point (3,π/2) lies on the graph of r=3 cos 2θ.

검증된 단계별 안내

Recall that the graph is given in polar coordinates by the equation \(r = 3 \cos 2\theta\).

To check if the point \((3, \frac{\pi}{2})\) lies on the graph, substitute \(\theta = \frac{\pi}{2}\) into the equation to find the corresponding \(r\) value.

Calculate \(r\) by evaluating \(3 \cos \left( 2 \times \frac{\pi}{2} \right) = 3 \cos \pi\).

Recall that \(\cos \pi = -1\), so \(r = 3 \times (-1) = -3\).

Since the given point has \(r = 3\) but the equation yields \(r = -3\) at \(\theta = \frac{\pi}{2}\), the point \((3, \frac{\pi}{2})\) does not lie on the graph.

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주요 개념

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Polar Coordinates

Polar coordinates represent points in the plane using a radius and an angle (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. Understanding how to interpret and plot points in polar form is essential for analyzing whether a point lies on a given polar curve.

Polar Equations and Graphs

A polar equation expresses the radius r as a function of the angle θ, such as r = 3 cos 2θ. To determine if a point lies on the graph, substitute the given θ into the equation and check if the resulting r matches the point's radius. This process verifies the point's membership on the curve.

Trigonometric Function Evaluation

Evaluating trigonometric functions like cosine at specific angles is crucial for solving polar equations. For example, calculating cos(2θ) when θ = π/2 requires knowledge of angle doubling and cosine values, which helps determine the correct radius and verify if the point satisfies the equation.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. d. The point (3,π/2) lies on the graph of r=3 cos 2θ.