On this page we discuss calculating slope and tangent lines of parametric equations.
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To find the equation of a tangent line to a graph given by a set of parametric equations, we need to be able to find the slope by calculating the derivative \( dy/dx \) using the parametric derivative
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \text{ where } dx/dt \neq 0 \]
For horizontal tangent lines, the slope \(dy/dx\) is zero, so we need \( dy/dt = 0 \) and \( dx/dt \neq 0 \). For vertical tangent lines, the slope is undefined, which means that \( dx/dt = 0 \) when \( dy/dt \neq 0 \). In the case where both \( dx/dt = 0\) and \( dy/dt = 0 \) at the same point, we need to handle that case separately, since nothing can be concluded from \( dy/dx = 0/0 \), which is indeterminate.
Once you have found the slope, you can easily find the equation of a tangent line. Go to the Tangent Lines page for more information.
Practice
Unless otherwise instructed, find the equation of the tangent line to these parametric curves at the given points.
\( x=4t \), \( y=3t^2+2 \); \(t=2\).
Problem Statement 

Find the equation of the tangent line to the parametric curve \( x=4t \), \( y=3t^2+2 \) at \(t=2\).
Solution 

video by The Organic Chemistry Tutor 

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\( x=t^25 \), \( y=t^24t \); \( (4,3) \)
Problem Statement 

Find the equation of the tangent line to the parametric curve \( x=t^25 \), \( y=t^24t \) at \( (4,3) \).
Solution 

video by The Organic Chemistry Tutor 

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\( x=8\cos\theta \), \( y=6\sin\theta \); \( \theta = \pi/3 \)
Problem Statement 

Find the equation of the tangent line to the parametric curve \( x=8\cos\theta \), \( y=6\sin\theta \) at \( \theta = \pi/3 \).
Solution 

video by The Organic Chemistry Tutor 

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\(x=2t^2+1\), \(y=3t^3+2\); \(t=1\).
Problem Statement 

Find the equation of the tangent line to the parametric curve \(x=2t^2+1\), \(y=3t^3+2\) at \(t=1\).
Solution 

video by Krista King Math 

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\(x=3(t\sin(t))\), \(y=3(1\cos(t))\); \(t=\pi/2\).
Problem Statement 

Find the equation of the tangent line to the parametric curve \(x=3(t\sin(t))\), \(y=3(1\cos(t))\) at \(t=\pi/2\).
Solution 

video by Krista King Math 

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\(x=t\cos(t)\), \(y=t\sin(t)\); \(t=\pi\)
Problem Statement 

Find the equation of the tangent line to the parametric curve \(x=t\cos(t)\), \(y=t\sin(t)\) at \(t=\pi\).
Solution 

video by Krista King Math 

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\(x=3t^2t\), \(y=\sqrt{t}\); \(t=4\).
Problem Statement 

Find the equation of the tangent line to the parametric curve \(x=3t^2t\), \(y=\sqrt{t}\) at \(t=4\).
Solution 

video by PatrickJMT 

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The Unit Circle
The Unit Circle [wikipedia]
Basic Trig Identities
Set 1  basic identities  

\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\) 
\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\) 
\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\) 
\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\) 
Set 2  squared identities  

\( \sin^2t + \cos^2t = 1\) 
\( 1 + \tan^2t = \sec^2t\) 
\( 1 + \cot^2t = \csc^2t\) 
Set 3  doubleangle formulas  

\( \sin(2t) = 2\sin(t)\cos(t)\) 
\(\displaystyle{ \cos(2t) = \cos^2(t)  \sin^2(t) }\) 
Set 4  halfangle formulas  

\(\displaystyle{ \sin^2(t) = \frac{1\cos(2t)}{2} }\) 
\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\) 
Trig Derivatives
\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\) 
\(\displaystyle{ \frac{d[\cos(t)]}{dt} = \sin(t) }\)  
\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\) 
\(\displaystyle{ \frac{d[\cot(t)]}{dt} = \csc^2(t) }\)  
\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\) 
\(\displaystyle{ \frac{d[\csc(t)]}{dt} = \csc(t)\cot(t) }\) 
Inverse Trig Derivatives
\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1t^2}} }\) 
\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = \frac{1}{\sqrt{1t^2}} }\)  
\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\) 
\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = \frac{1}{1+t^2} }\)  
\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 1}} }\) 
\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 1}} }\) 
Trig Integrals
\(\int{\sin(x)~dx} = \cos(x)+C\) 
\(\int{\cos(x)~dx} = \sin(x)+C\)  
\(\int{\tan(x)~dx} = \ln\abs{\cos(x)}+C\) 
\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)  
\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\) 
\(\int{\csc(x)~dx} = \) \( \ln\abs{\csc(x)+\cot(x)}+C\) 
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Practice Instructions
Unless otherwise instructed, find the equation of the tangent line to these parametric curves at the given points.