The slope of the tangent line to a polar curve r = f(θ) at a given angle. 60 problems with worked solutions; in 60 of them every
equation is proved by a computer algebra system.
Try each one before opening the solution. Every line is labeled with what checked it. How we verify
Find the slope of the tangent line to \( \displaystyle r = \cos{\left(\theta \right)} + 1 \) at \( \displaystyle \theta = \frac{\pi}{3} \).
Show the first step
Show the whole solution
With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).
Reviewed
\[ \frac{d}{d \theta} \left(\cos{\left(\theta \right)} + 1\right) \cos{\left(\theta \right)} = \left(- \cos{\left(\theta \right)} - 1\right) \sin{\left(\theta \right)} - \sin{\left(\theta \right)} \cos{\left(\theta \right)} \]
dx/dθ.✓ Proved
\[ \frac{d}{d \theta} \left(\cos{\left(\theta \right)} + 1\right) \sin{\left(\theta \right)} = \left(\cos{\left(\theta \right)} + 1\right) \cos{\left(\theta \right)} - \sin^{2}{\left(\theta \right)} \]
dy/dθ.✓ Proved
\[ \frac{\left. \left(\cos{\left(\theta \right)} + 1\right) \cos{\left(\theta \right)} - \sin^{2}{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{3} }}}{\left. \left(- \cos{\left(\theta \right)} - 1\right) \sin{\left(\theta \right)} - \sin{\left(\theta \right)} \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{3} }}} = 0 \]
At θ = pi/3.✓ Proved
Answer \( 0 \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the slope of the tangent line to \( \displaystyle r = 2 \cos{\left(\theta \right)} \) at \( \displaystyle \theta = \frac{\pi}{6} \).
Show the first step
Show the whole solution
With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).
Reviewed
\[ \frac{d}{d \theta} 2 \cos^{2}{\left(\theta \right)} = - 4 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \]
dx/dθ.✓ Proved
\[ \frac{d}{d \theta} 2 \sin{\left(\theta \right)} \cos{\left(\theta \right)} = - 2 \sin^{2}{\left(\theta \right)} + 2 \cos^{2}{\left(\theta \right)} \]
dy/dθ.✓ Proved
\[ \frac{\left. - 2 \sin^{2}{\left(\theta \right)} + 2 \cos^{2}{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{6} }}}{\left. - 4 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{6} }}} = - \frac{\sqrt{3}}{3} \]
At θ = pi/6.✓ Proved
Answer \( - \frac{\sqrt{3}}{3} \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the slope of the tangent line to \( \displaystyle r = 4 \cos{\left(2 \theta \right)} \) at \( \displaystyle \theta = \frac{\pi}{4} \).
Show the first step
Show the whole solution
With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).
Reviewed
\[ \frac{d}{d \theta} 4 \cos{\left(\theta \right)} \cos{\left(2 \theta \right)} = - 4 \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} - 8 \sin{\left(2 \theta \right)} \cos{\left(\theta \right)} \]
dx/dθ.✓ Proved
\[ \frac{d}{d \theta} 4 \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} = - 8 \sin{\left(\theta \right)} \sin{\left(2 \theta \right)} + 4 \cos{\left(\theta \right)} \cos{\left(2 \theta \right)} \]
dy/dθ.✓ Proved
\[ \frac{\left. - 8 \sin{\left(\theta \right)} \sin{\left(2 \theta \right)} + 4 \cos{\left(\theta \right)} \cos{\left(2 \theta \right)} \right|_{\substack{ \theta=\frac{\pi}{4} }}}{\left. - 4 \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} - 8 \sin{\left(2 \theta \right)} \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{4} }}} = 1 \]
At θ = pi/4.✓ Proved
Answer \( 1 \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the slope of the tangent line to \( \displaystyle r = \theta \) at \( \displaystyle \theta = \frac{\pi}{6} \).
Show the first step
Show the whole solution
With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).
Reviewed
\[ \frac{d}{d \theta} \theta \cos{\left(\theta \right)} = - \theta \sin{\left(\theta \right)} + \cos{\left(\theta \right)} \]
dx/dθ.✓ Proved
\[ \frac{d}{d \theta} \theta \sin{\left(\theta \right)} = \theta \cos{\left(\theta \right)} + \sin{\left(\theta \right)} \]
dy/dθ.✓ Proved
\[ \frac{\left. \theta \cos{\left(\theta \right)} + \sin{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{6} }}}{\left. - \theta \sin{\left(\theta \right)} + \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{6} }}} = \frac{\sqrt{3} \pi + 6}{- \pi + 6 \sqrt{3}} \]
At θ = pi/6.✓ Proved
Answer \( \frac{\sqrt{3} \pi + 6}{- \pi + 6 \sqrt{3}} \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the slope of the tangent line to \( \displaystyle r = 2 \) at \( \displaystyle \theta = \frac{\pi}{2} \).
Show the first step
Show the whole solution
With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).
Reviewed
\[ \frac{d}{d \theta} 2 \cos{\left(\theta \right)} = - 2 \sin{\left(\theta \right)} \]
dx/dθ.✓ Proved
\[ \frac{d}{d \theta} 2 \sin{\left(\theta \right)} = 2 \cos{\left(\theta \right)} \]
dy/dθ.✓ Proved
\[ \frac{\left. 2 \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{2} }}}{\left. - 2 \sin{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{2} }}} = 0 \]
At θ = pi/2.✓ Proved
Answer \( 0 \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the slope of the tangent line to \( \displaystyle r = 2 \sin{\left(\theta \right)} \) at \( \displaystyle \theta = \frac{\pi}{2} \).
Show the first step
Show the whole solution
With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).
Reviewed
\[ \frac{d}{d \theta} 2 \sin{\left(\theta \right)} \cos{\left(\theta \right)} = - 2 \sin^{2}{\left(\theta \right)} + 2 \cos^{2}{\left(\theta \right)} \]
dx/dθ.✓ Proved
\[ \frac{d}{d \theta} 2 \sin^{2}{\left(\theta \right)} = 4 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \]
dy/dθ.✓ Proved
\[ \frac{\left. 4 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{2} }}}{\left. - 2 \sin^{2}{\left(\theta \right)} + 2 \cos^{2}{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{2} }}} = 0 \]
At θ = pi/2.✓ Proved
Answer \( 0 \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the slope of the tangent line to \( \displaystyle r = 2 \cos{\left(\theta \right)} + 2 \) at \( \displaystyle \theta = \frac{\pi}{2} \).
Show the first step
Show the whole solution
With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).
Reviewed
\[ \frac{d}{d \theta} \left(2 \cos{\left(\theta \right)} + 2\right) \cos{\left(\theta \right)} = \left(- 2 \cos{\left(\theta \right)} - 2\right) \sin{\left(\theta \right)} - 2 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \]
dx/dθ.✓ Proved
\[ \frac{d}{d \theta} \left(2 \cos{\left(\theta \right)} + 2\right) \sin{\left(\theta \right)} = \left(2 \cos{\left(\theta \right)} + 2\right) \cos{\left(\theta \right)} - 2 \sin^{2}{\left(\theta \right)} \]
dy/dθ.✓ Proved
\[ \frac{\left. \left(2 \cos{\left(\theta \right)} + 2\right) \cos{\left(\theta \right)} - 2 \sin^{2}{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{2} }}}{\left. \left(- 2 \cos{\left(\theta \right)} - 2\right) \sin{\left(\theta \right)} - 2 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{2} }}} = 1 \]
At θ = pi/2.✓ Proved
Answer \( 1 \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the slope of the tangent line to \( \displaystyle r = 4 \) at \( \displaystyle \theta = \frac{2 \pi}{3} \).
Show the first step
Show the whole solution
With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).
Reviewed
\[ \frac{d}{d \theta} 4 \cos{\left(\theta \right)} = - 4 \sin{\left(\theta \right)} \]
dx/dθ.✓ Proved
\[ \frac{d}{d \theta} 4 \sin{\left(\theta \right)} = 4 \cos{\left(\theta \right)} \]
dy/dθ.✓ Proved
\[ \frac{\left. 4 \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{2 \pi}{3} }}}{\left. - 4 \sin{\left(\theta \right)} \right|_{\substack{ \theta=\frac{2 \pi}{3} }}} = \frac{\sqrt{3}}{3} \]
At θ = 2*pi/3.✓ Proved
Answer \( \frac{\sqrt{3}}{3} \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the slope of the tangent line to \( \displaystyle r = 4 \cos{\left(\theta \right)} \) at \( \displaystyle \theta = \frac{2 \pi}{3} \).
Show the first step
Show the whole solution
With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).
Reviewed
\[ \frac{d}{d \theta} 4 \cos^{2}{\left(\theta \right)} = - 8 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \]
dx/dθ.✓ Proved
\[ \frac{d}{d \theta} 4 \sin{\left(\theta \right)} \cos{\left(\theta \right)} = - 4 \sin^{2}{\left(\theta \right)} + 4 \cos^{2}{\left(\theta \right)} \]
dy/dθ.✓ Proved
\[ \frac{\left. - 4 \sin^{2}{\left(\theta \right)} + 4 \cos^{2}{\left(\theta \right)} \right|_{\substack{ \theta=\frac{2 \pi}{3} }}}{\left. - 8 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{2 \pi}{3} }}} = - \frac{\sqrt{3}}{3} \]
At θ = 2*pi/3.✓ Proved
Answer \( - \frac{\sqrt{3}}{3} \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the slope of the tangent line to \( \displaystyle r = 1 - \sin{\left(\theta \right)} \) at \( \displaystyle \theta = \frac{2 \pi}{3} \).
Show the first step
Show the whole solution
With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).
Reviewed
\[ \frac{d}{d \theta} \left(1 - \sin{\left(\theta \right)}\right) \cos{\left(\theta \right)} = \left(\sin{\left(\theta \right)} - 1\right) \sin{\left(\theta \right)} - \cos^{2}{\left(\theta \right)} \]
dx/dθ.✓ Proved
\[ \frac{d}{d \theta} \left(1 - \sin{\left(\theta \right)}\right) \sin{\left(\theta \right)} = \left(1 - \sin{\left(\theta \right)}\right) \cos{\left(\theta \right)} - \sin{\left(\theta \right)} \cos{\left(\theta \right)} \]
dy/dθ.✓ Proved
\[ \frac{\left. \left(1 - \sin{\left(\theta \right)}\right) \cos{\left(\theta \right)} - \sin{\left(\theta \right)} \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{2 \pi}{3} }}}{\left. \left(\sin{\left(\theta \right)} - 1\right) \sin{\left(\theta \right)} - \cos^{2}{\left(\theta \right)} \right|_{\substack{ \theta=\frac{2 \pi}{3} }}} = -1 \]
At θ = 2*pi/3.✓ Proved
Answer \( -1 \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the slope of the tangent line to \( \displaystyle r = 3 - 3 \sin{\left(\theta \right)} \) at \( \displaystyle \theta = \frac{\pi}{3} \).
Show the first step
Show the whole solution
With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).
Reviewed
\[ \frac{d}{d \theta} \left(3 - 3 \sin{\left(\theta \right)}\right) \cos{\left(\theta \right)} = \left(3 \sin{\left(\theta \right)} - 3\right) \sin{\left(\theta \right)} - 3 \cos^{2}{\left(\theta \right)} \]
dx/dθ.✓ Proved
\[ \frac{d}{d \theta} \left(3 - 3 \sin{\left(\theta \right)}\right) \sin{\left(\theta \right)} = \left(3 - 3 \sin{\left(\theta \right)}\right) \cos{\left(\theta \right)} - 3 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \]
dy/dθ.✓ Proved
\[ \frac{\left. \left(3 - 3 \sin{\left(\theta \right)}\right) \cos{\left(\theta \right)} - 3 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{3} }}}{\left. \left(3 \sin{\left(\theta \right)} - 3\right) \sin{\left(\theta \right)} - 3 \cos^{2}{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{3} }}} = 1 \]
At θ = pi/3.✓ Proved
Answer \( 1 \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the slope of the tangent line to \( \displaystyle r = 2 \cos{\left(2 \theta \right)} \) at \( \displaystyle \theta = \frac{2 \pi}{3} \).
Show the first step
Show the whole solution
With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).
Reviewed
\[ \frac{d}{d \theta} 2 \cos{\left(\theta \right)} \cos{\left(2 \theta \right)} = - 2 \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} - 4 \sin{\left(2 \theta \right)} \cos{\left(\theta \right)} \]
dx/dθ.✓ Proved
\[ \frac{d}{d \theta} 2 \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} = - 4 \sin{\left(\theta \right)} \sin{\left(2 \theta \right)} + 2 \cos{\left(\theta \right)} \cos{\left(2 \theta \right)} \]
dy/dθ.✓ Proved
\[ \frac{\left. - 4 \sin{\left(\theta \right)} \sin{\left(2 \theta \right)} + 2 \cos{\left(\theta \right)} \cos{\left(2 \theta \right)} \right|_{\substack{ \theta=\frac{2 \pi}{3} }}}{\left. - 2 \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} - 4 \sin{\left(2 \theta \right)} \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{2 \pi}{3} }}} = - \frac{7 \sqrt{3}}{3} \]
At θ = 2*pi/3.✓ Proved
Answer \( - \frac{7 \sqrt{3}}{3} \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the slope of the tangent line to \( \displaystyle r = \cos{\left(2 \theta \right)} \) at \( \displaystyle \theta = \frac{\pi}{3} \).
Show the first step
Show the whole solution
With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).
Reviewed
\[ \frac{d}{d \theta} \cos{\left(\theta \right)} \cos{\left(2 \theta \right)} = - \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} - 2 \sin{\left(2 \theta \right)} \cos{\left(\theta \right)} \]
dx/dθ.✓ Proved
\[ \frac{d}{d \theta} \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} = - 2 \sin{\left(\theta \right)} \sin{\left(2 \theta \right)} + \cos{\left(\theta \right)} \cos{\left(2 \theta \right)} \]
dy/dθ.✓ Proved
\[ \frac{\left. - 2 \sin{\left(\theta \right)} \sin{\left(2 \theta \right)} + \cos{\left(\theta \right)} \cos{\left(2 \theta \right)} \right|_{\substack{ \theta=\frac{\pi}{3} }}}{\left. - \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} - 2 \sin{\left(2 \theta \right)} \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{3} }}} = \frac{7 \sqrt{3}}{3} \]
At θ = pi/3.✓ Proved
Answer \( \frac{7 \sqrt{3}}{3} \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the slope of the tangent line to \( \displaystyle r = 1 - \sin{\left(\theta \right)} \) at \( \displaystyle \theta = \frac{\pi}{4} \).
Show the first step
Show the whole solution
With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).
Reviewed
\[ \frac{d}{d \theta} \left(1 - \sin{\left(\theta \right)}\right) \cos{\left(\theta \right)} = \left(\sin{\left(\theta \right)} - 1\right) \sin{\left(\theta \right)} - \cos^{2}{\left(\theta \right)} \]
dx/dθ.✓ Proved
\[ \frac{d}{d \theta} \left(1 - \sin{\left(\theta \right)}\right) \sin{\left(\theta \right)} = \left(1 - \sin{\left(\theta \right)}\right) \cos{\left(\theta \right)} - \sin{\left(\theta \right)} \cos{\left(\theta \right)} \]
dy/dθ.✓ Proved
\[ \frac{\left. \left(1 - \sin{\left(\theta \right)}\right) \cos{\left(\theta \right)} - \sin{\left(\theta \right)} \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{4} }}}{\left. \left(\sin{\left(\theta \right)} - 1\right) \sin{\left(\theta \right)} - \cos^{2}{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{4} }}} = -1 + \sqrt{2} \]
At θ = pi/4.✓ Proved
Answer \( -1 + \sqrt{2} \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the slope of the tangent line to \( \displaystyle r = 4 \sin{\left(\theta \right)} \) at \( \displaystyle \theta = \frac{2 \pi}{3} \).
Show the first step
Show the whole solution
With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).
Reviewed
\[ \frac{d}{d \theta} 4 \sin{\left(\theta \right)} \cos{\left(\theta \right)} = - 4 \sin^{2}{\left(\theta \right)} + 4 \cos^{2}{\left(\theta \right)} \]
dx/dθ.✓ Proved
\[ \frac{d}{d \theta} 4 \sin^{2}{\left(\theta \right)} = 8 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \]
dy/dθ.✓ Proved
\[ \frac{\left. 8 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{2 \pi}{3} }}}{\left. - 4 \sin^{2}{\left(\theta \right)} + 4 \cos^{2}{\left(\theta \right)} \right|_{\substack{ \theta=\frac{2 \pi}{3} }}} = \sqrt{3} \]
At θ = 2*pi/3.✓ Proved
Answer \( \sqrt{3} \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the slope of the tangent line to \( \displaystyle r = \cos{\left(2 \theta \right)} \) at \( \displaystyle \theta = \frac{2 \pi}{3} \).
Show the first step
Show the whole solution
With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).
Reviewed
\[ \frac{d}{d \theta} \cos{\left(\theta \right)} \cos{\left(2 \theta \right)} = - \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} - 2 \sin{\left(2 \theta \right)} \cos{\left(\theta \right)} \]
dx/dθ.✓ Proved
\[ \frac{d}{d \theta} \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} = - 2 \sin{\left(\theta \right)} \sin{\left(2 \theta \right)} + \cos{\left(\theta \right)} \cos{\left(2 \theta \right)} \]
dy/dθ.✓ Proved
\[ \frac{\left. - 2 \sin{\left(\theta \right)} \sin{\left(2 \theta \right)} + \cos{\left(\theta \right)} \cos{\left(2 \theta \right)} \right|_{\substack{ \theta=\frac{2 \pi}{3} }}}{\left. - \sin{\left(\theta \right)} \cos{\left(2 \theta \right)} - 2 \sin{\left(2 \theta \right)} \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{2 \pi}{3} }}} = - \frac{7 \sqrt{3}}{3} \]
At θ = 2*pi/3.✓ Proved
Answer \( - \frac{7 \sqrt{3}}{3} \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the slope of the tangent line to \( \displaystyle r = \cos{\left(\theta \right)} \) at \( \displaystyle \theta = \frac{\pi}{3} \).
Show the first step
Show the whole solution
With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).
Reviewed
\[ \frac{d}{d \theta} \cos^{2}{\left(\theta \right)} = - 2 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \]
dx/dθ.✓ Proved
\[ \frac{d}{d \theta} \sin{\left(\theta \right)} \cos{\left(\theta \right)} = - \sin^{2}{\left(\theta \right)} + \cos^{2}{\left(\theta \right)} \]
dy/dθ.✓ Proved
\[ \frac{\left. - \sin^{2}{\left(\theta \right)} + \cos^{2}{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{3} }}}{\left. - 2 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{3} }}} = \frac{\sqrt{3}}{3} \]
At θ = pi/3.✓ Proved
Answer \( \frac{\sqrt{3}}{3} \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the slope of the tangent line to \( \displaystyle r = \sin{\left(\theta \right)} \) at \( \displaystyle \theta = \frac{\pi}{2} \).
Show the first step
Show the whole solution
With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).
Reviewed
\[ \frac{d}{d \theta} \sin{\left(\theta \right)} \cos{\left(\theta \right)} = - \sin^{2}{\left(\theta \right)} + \cos^{2}{\left(\theta \right)} \]
dx/dθ.✓ Proved
\[ \frac{d}{d \theta} \sin^{2}{\left(\theta \right)} = 2 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \]
dy/dθ.✓ Proved
\[ \frac{\left. 2 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{2} }}}{\left. - \sin^{2}{\left(\theta \right)} + \cos^{2}{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{2} }}} = 0 \]
At θ = pi/2.✓ Proved
Answer \( 0 \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the slope of the tangent line to \( \displaystyle r = \sin{\left(\theta \right)} \) at \( \displaystyle \theta = \frac{2 \pi}{3} \).
Show the first step
Show the whole solution
With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).
Reviewed
\[ \frac{d}{d \theta} \sin{\left(\theta \right)} \cos{\left(\theta \right)} = - \sin^{2}{\left(\theta \right)} + \cos^{2}{\left(\theta \right)} \]
dx/dθ.✓ Proved
\[ \frac{d}{d \theta} \sin^{2}{\left(\theta \right)} = 2 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \]
dy/dθ.✓ Proved
\[ \frac{\left. 2 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{2 \pi}{3} }}}{\left. - \sin^{2}{\left(\theta \right)} + \cos^{2}{\left(\theta \right)} \right|_{\substack{ \theta=\frac{2 \pi}{3} }}} = \sqrt{3} \]
At θ = 2*pi/3.✓ Proved
Answer \( \sqrt{3} \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the slope of the tangent line to \( \displaystyle r = 2 - 2 \sin{\left(\theta \right)} \) at \( \displaystyle \theta = \frac{2 \pi}{3} \).
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With x = r cos θ and y = r sin θ, dy/dx = (dy/dθ)/(dx/dθ).
Reviewed
\[ \frac{d}{d \theta} \left(2 - 2 \sin{\left(\theta \right)}\right) \cos{\left(\theta \right)} = \left(2 \sin{\left(\theta \right)} - 2\right) \sin{\left(\theta \right)} - 2 \cos^{2}{\left(\theta \right)} \]
dx/dθ.✓ Proved
\[ \frac{d}{d \theta} \left(2 - 2 \sin{\left(\theta \right)}\right) \sin{\left(\theta \right)} = \left(2 - 2 \sin{\left(\theta \right)}\right) \cos{\left(\theta \right)} - 2 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \]
dy/dθ.✓ Proved
\[ \frac{\left. \left(2 - 2 \sin{\left(\theta \right)}\right) \cos{\left(\theta \right)} - 2 \sin{\left(\theta \right)} \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{2 \pi}{3} }}}{\left. \left(2 \sin{\left(\theta \right)} - 2\right) \sin{\left(\theta \right)} - 2 \cos^{2}{\left(\theta \right)} \right|_{\substack{ \theta=\frac{2 \pi}{3} }}} = -1 \]
At θ = 2*pi/3.✓ Proved
Answer \( -1 \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify