Question

Find the equation (in terms of (x) and (y)) of the tangent line to the curve (r=5 sin 2 θ) at (θ=π/ 4)?

Answer 1

The equation of the tangent line to the curve r=5 sin 2θ at θ=π/4 requires conversion from polar to Cartesian coordinates and finding the slope of the tangent line to construct the line equation.

Step-by-step explanation:

To find the equation of the tangent line to the curve given by r=5 sin 2θ at θ=π/4, you need to convert the polar coordinates to Cartesian coordinates and then calculate the slope of the tangent line.

Step 1: Convert to Cartesian coordinates

At θ=π/4, the polar coordinates (r, θ) can be converted to (x, y) using:

y = r × sin θ

Step 2: Find the slope of the tangent line

The slope of the tangent line (dy/dx) at the curve r=5 sin 2θ can be found using the derivative of r with respect to θ and using the chain rule for derivatives.

Step 3: Write the equation of the tangent line

The equation of the tangent line at a point (x, y) is given by y - y1 = m(x - x1), where (x1, y1) are the coordinates of the point of tangency and m is the slope found in Step 2.

Answer 2

To find the tangent line to the given polar curve at θ=π/ 4, convert the polar coordinates to Cartesian, calculate the slope of the tangent line using differentiation, and then write the equation of the tangent in Cartesian form, which is y = 5/√2.

Step-by-step explanation:

To find the equation of the tangent line to the curve r=5 sin 2θ at θ=π/ 4, we first need to convert the polar equation to Cartesian form using the relationships x = r × cos(θ) and y = r × sin(θ). At θ=π/ 4, r equals 5 since sin(2 × π/ 4) = sin(π/ 2) = 1. Thus, the point of tangency in Cartesian coordinates is (x, y) = (5cos(π/ 4), 5sin(π/ 4)), which simplifies to (5/√2, 5/√2).

To find the slope of the tangent line, we differentiate the polar equation with respect to θ to find dr/dθ and use the formula dy/dx = (dr/dθ × sinθ + r × cosθ)/(dr/dθ × cosθ - r × sinθ). After differentiation, we get dr/dθ = 10cos(2θ). Substituting θ = π/ 4 and r = 5, we find that the slope m at our point is (10cos(π/ 2) × sin(π/ 4) + 5 × cos(π/ 4))/(10cos(π/ 2) × cos(π/ 4) - 5 × sin(π/ 4)) = 0, since cos(π/ 2) = 0. Therefore, the tangent line is horizontal, and its equation is simply y = 5/√2.