Question

curve \(\mathrm{y}=(3 / 2) \sin 2 \theta, \mathrm{x}=\mathrm{e}^{\theta} \cdot \sin \theta, 0<2\) for which value of \(\theta\) tangent is parallel to X-axis? (a) 0 (b) \((\pi / 2)\) (c) \((\pi / 4)\) (d) \((\pi / 6)\)

Short answer

The value of θ for which the tangent is parallel to the X-axis is \(\frac{\pi}{4}\), which is option (c).

Step-by-step solution

1. Compute derivatives for x and y

First, let's differentiate x and y with respect to θ: Given: \(y = \frac{3}{2} \sin{2\theta}\) and \(x = e^\theta \sin\theta\) Derivative of y with respect to θ: \(\frac{dy}{d\theta} = \frac{3}{2} (2\cos2\theta)\) Derivative of x with respect to θ: \(\frac{dx}{d\theta} = e^\theta\sin{\theta} + e^\theta\cos{\theta}\)

2. Compute dy/dx

Now, let's compute dy/dx: \(\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}\) \(\frac{dy}{dx} = \frac{3\cos{2\theta}}{2(e^\theta\sin{\theta} + e^\theta\cos{\theta})}\)

3. Set dy/dx to 0 and solve for θ

Since the tangent is parallel to the X-axis, the slope of the tangent should be equal to 0. Hence, we have to set dy/dx equal to 0 and solve for θ: \(\frac{3\cos{2\theta}}{2(e^\theta\sin{\theta} + e^\theta\cos{\theta})} = 0\) To satisfy the equation, the numerator must be equal to 0. Therefore: \(3\cos{2\theta} = 0\)

4. Solve for θ and check the given options

Now let's solve for θ and determine which of the given options satisfies the equation: \(\cos{2\theta} = 0\) \(2\theta = \frac{\pi}{2} + n\pi\) where n is an integer. \(\theta = \frac{\pi}{4} + \frac{n\pi}{2}\) Now let's check which of the given options satisfy the above equation. (a) 0 - No (b) \(\frac{\pi}{2}\) - No (c) \(\frac{\pi}{4}\) - Yes (d) \(\frac{\pi}{6}\) - No Hence, the value of θ for which the tangent is parallel to the X-axis is \(\boxed{\frac{\pi}{4}}\), which is option (c).

Key concepts

Calculus

Calculus is a branch of mathematics focused on changes. It primarily deals with two concepts: differentiation and integration. Differentiation helps find the rate of change of a quantity, while integration is used for finding areas under curves or the accumulation of quantities. Understanding calculus is essential for analyzing curves and motion, predicting trajectories, and solving practical problems. In this exercise, we use differentiation to find when the tangent to a curve is parallel to the X-axis. This involves calculating derivatives, a fundamental aspect of calculus.

Tangent to a Curve

The tangent to a curve at any point is a straight line that just touches the curve at that point. It is like a line that gently brushes by the curve.A tangent tells us the immediate direction of the curve at that point, providing insights about its slope and orientation.If the tangent is parallel to the X-axis, it means the slope is zero. This indicates that the curve has a horizontal flat touch at that point.In the solution, setting the derivative \( \frac{dy}{dx} = 0 \) helped identify when this condition occurs. It ensures the tangent has no vertical component.

Derivatives

Derivatives are central to calculus. They provide the slope of a curve at any point and represent rates of change.By taking the derivative of a function, we can determine how fast or slow a value is changing.This exercise asked for the derivatives of both x and y with respect to \( \theta \).

• \( \frac{dy}{d\theta} = \frac{3}{2}(2\cos{2\theta}) \)
• \( \frac{dx}{d\theta} = e^{\theta}\sin\theta + e^{\theta}\cos\theta \)

The calculation of these derivatives led to the fundamental expression \( \frac{dy}{dx} \), the tool for finding when the slope is zero.

Trigonometry

Trigonometry focuses on angles, triangles, and periodic functions related to sine, cosine, and tangent.In our exercise, trigonometry helps understand how these trigonometric functions behave when the curve, expressed as a function of \( \theta \), changes.We used the identity \( \cos{2\theta} = 0 \) to find when the slope is zero.Some trigonometric identities, such as \( \sin^2{\theta} + \cos^2{\theta} = 1 \), are invaluable in solving such equations.Here, understanding trigonometric transformations allowed the keystone simplification that solved for \( \theta = \frac{\pi}{4} + \frac{n\pi}{2} \). This uncovered when the tangent to the curve meets our desired condition of being parallel to the X-axis.