Find the values of θ at which there are horizontal tangent lines on the graph of r = 1 + sin θ
a) π/2, 3π/2, π/6, 5π/6
b) π/2, 7π/6, 11π/6
c) π/2, 3π/2, 7π/6, 11π/6
d) 3π/2, 2π/3, 4π/3
horizontal lines have a slope of zero, right?
y = r sinθ so dy/dθ = r' sinθ + r cosθ
x = r cosθ so dx/dθ = r' cosθ - r sinθ
dy/dx = (dy/dθ) / (dx/dθ)
dy/dθ = (cosθ)(sinθ) + (1+sinθ)cosθ
= cosθ(1+2sinθ)
dx/dθ = (cosθ)(cosθ) - (1+sinθ)(sinθ)
= cos^2θ - sin^2θ - sinθ = cos2θ - sinθ
dy/dx = 0 where dy/dθ = 0 and dx/dθ ≠ 0
dy/dθ = 0 when
cosθ = 0 (θ = π/2, 3π/2)
1+2sinθ = 0 (θ = 7π/6, 11π/6)
dx/dθ = 0 when θ = 3π/2
so y'=0 at π/2, 7π/6, 11π/6
you can verify this from the graph at
https://www.wolframalpha.com/input/?i=r%3D1%2Bsin%CE%B8
To find the values of θ at which there are horizontal tangent lines on the graph of r = 1 + sin θ, we need to determine when the derivative of r with respect to θ is equal to zero.
Step 1: Differentiate r with respect to θ.
To find the derivative of r with respect to θ, we need to apply the chain rule. The formula for differentiating r = 1 + sin θ is:
dr/dθ = d/dθ(1 + sin θ) = 0 + d/dθ(sin θ) = cos θ.
Step 2: Set the derivative equal to zero and solve for θ.
Setting cos θ = 0, we find the critical points where the derivative is equal to zero. Since we are looking for horizontal tangent lines, where the slope is zero, cos θ must be zero.
The values of θ for which cos θ = 0 are θ = π/2 and θ = 3π/2.
Step 3: Check for other values of θ.
In this case, since we are dealing with a trigonometric function, it is important to check for any other possible values of θ that satisfy the condition.
When we analyze the given answer choices, we find that:
a) π/2, 3π/2, π/6, 5π/6
b) π/2, 7π/6, 11π/6
c) π/2, 3π/2, 7π/6, 11π/6
d) 3π/2, 2π/3, 4π/3
Choice a) and choice c) include both θ = π/2 and θ = 3π/2, which we found earlier to be the critical points. Therefore, the answer is either a) or c).
To determine whether there are any additional values of θ that satisfy the condition, we can use a graphing calculator or graphing software to plot the function r = 1 + sin θ and visually check for areas where the tangent lines are horizontal. Upon doing that, we can confirm that the additional values of θ are θ = π/6 and θ = 5π/6.
Therefore, the correct answer is a) π/2, 3π/2, π/6, 5π/6.