This page covers how to solve linear systems of equations using determinants.
We can use matrices and determinants to solve systems of linear equations using a technique called Cramers Rule. We will show how to do this with 2 equations and 2 unknowns. However, the concept can be extended to higher order systems.
Problem
We are given a system of linear equations of the form
\(
\begin{array}{ccccc}
ax & + & by & = & z_0 \\
cx & + & dy & = & z_1
\end{array}
\)
where \( a, b, c, d, z_0, z_1 \) are all real constants and the variables are \( x\) and \(y \).
We need to find what values of \(x\) and \(y\) solve this equation. There are three possible cases.
1. \(x\) and \(y\) are real, unique and not equal.
2. \(x\) and \(y\) are real and equal.
3. \(x\) and \(y\) are complex.
The key to determining which case holds is to look at the determinant of the coefficient matrix, i.e.
\(
\begin{vmatrix}
a & b \\
c & d
\end{vmatrix}
\)
If this determinant is zero, then we cannot use this technique and either case 2 or 3 hold. If this case is nonzero, then case 1 holds and we can solve this problem. Let's call the coefficient matrix \(A\) and so
\(
\abs{A} =
\begin{vmatrix}
a & b \\
c & d
\end{vmatrix}
\)
Solution
As long as the determinant of the coefficient matrix is NOT zero, we can solve this system of equations and the values are given by
\(\displaystyle{
x = \frac{
\begin{vmatrix}
z_0 & b \\
z_1 & d
\end{vmatrix}
}{\abs{A}}
}\)
and
\(\displaystyle{
y = \frac{
\begin{vmatrix}
a & z_0 \\
c & z_1
\end{vmatrix}
}{\abs{A}}
}\)
In each case above, notice that we have replaced the column of matrix \(A\) corresponding to the variable we are calculating with the \(z\) constants.
Here are a couple of videos with examples.
video by PatrickJMT 

video by PatrickJMT 

The best way to learn Cramer's Rule is by watching someone work specific problems and then working plenty on your own. To get started, here is a video showing, in general, how to use Cramer's Rule, then he does a specific example. After this video, you should be able to work problems on your own.
video by Thinkwell 

Here is a video showing a proof or justification for Cramer's Rule.
video by PatrickJMT 

Important!
Although most of the solutions below do not show the work, it is important to check your work when using Cramer's Rule. As you saw from the videos and you will see in the solutions, it is very, very easy to make mistakes that propagate and give you all wrong answers. So it is important to plug your answers back into all of the equations to make sure you have the correct answers. Your work may be considered incomplete if you do not check your answers (depending on what your instructor expects). So just get used to always check your answers. It could be the difference between a whole letter grade and it takes only a few seconds.
Okay, now you are ready for some practice problems.
Practice
Solve these linear systems using Cramer's Rule.
Let's start with some 2x2 systems
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x+4y=14; \) \( 2x3y=11 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x+4y=14; \) \( 2x3y=11 \)
Final Answer 

\(x=2, y=5\)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x+4y=14; \) \( 2x3y=11 \)
Solution 

video by Krista King Math 

Final Answer 

\(x=2, y=5\) 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 5x+7y=1; \) \( 6x+8y=1 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 5x+7y=1; \) \( 6x+8y=1 \)
Final Answer 

\(x=15/2, y=11/2\)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 5x+7y=1; \) \( 6x+8y=1 \)
Solution 

video by Thinkwell 

Final Answer 

\(x=15/2, y=11/2\) 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 4x2y=10; \) \( 3x5y=11 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 4x2y=10; \) \( 3x5y=11 \)
Final Answer 

\( x=2, y=1 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 4x2y=10; \) \( 3x5y=11 \)
Solution 

video by mattemath 

Final Answer 

\( x=2, y=1 \) 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( xy=3; \) \( x+4y=17 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( xy=3; \) \( x+4y=17 \)
Final Answer 

\( x=1, y=4 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( xy=3; \) \( x+4y=17 \)
Solution 

video by MIP4U 

Final Answer 

\( x=1, y=4 \) 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x3y=16;\) \( x+2y=1 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x3y=16;\) \( x+2y=1 \)
Final Answer 

\( x=5, y=2 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x3y=16;\) \( x+2y=1 \)
Solution 

video by MIP4U 

Final Answer 

\( x=5, y=2 \) 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + 5y = 26 \)
\( 5x  4y = 1 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + 5y = 26 \)
\( 5x  4y = 1 \)
Final Answer 

\( (3, 4) \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + 5y = 26 \)
\( 5x  4y = 1 \)
Solution 

Final Answer 

\( (3, 4) \) 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x  2y = 4 \)
\( 4x  y = 3 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x  2y = 4 \)
\( 4x  y = 3 \)
Final Answer 

\( (2, 5) \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x  2y = 4 \)
\( 4x  y = 3 \)
Solution 

Final Answer 

\( (2, 5) \) 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + 3y = 13 \)
\( 3x  5y = 9 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + 3y = 13 \)
\( 3x  5y = 9 \)
Final Answer 

\( (2, 3) \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + 3y = 13 \)
\( 3x  5y = 9 \)
Solution 

Final Answer 

\( (2, 3) \) 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x  7y = 1 \)
\( 3x + y = 13 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x  7y = 1 \)
\( 3x + y = 13 \)
Final Answer 

\( (4, 1) \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x  7y = 1 \)
\( 3x + y = 13 \)
Solution 

Final Answer 

\( (4, 1) \) 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x  2y = 3 \)
\( 3x + y = 5 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x  2y = 3 \)
\( 3x + y = 5 \)
Final Answer 

\( (1, 2) \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x  2y = 3 \)
\( 3x + y = 5 \)
Solution 

Final Answer 

\( (1, 2) \) 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x  y = 4 \)
\( 2x + y = 2 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x  y = 4 \)
\( 2x + y = 2 \)
Final Answer 

\( (2, 2) \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x  y = 4 \)
\( 2x + y = 2 \)
Solution 

Final Answer 

\( (2, 2) \) 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 4x  2y = 10 \)
\( 3x  5y = 11 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 4x  2y = 10 \)
\( 3x  5y = 11 \)
Final Answer 

\( (2, 1) \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 4x  2y = 10 \)
\( 3x  5y = 11 \)
Solution 

video by mattemath 

Final Answer 

\( (2, 1) \) 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x  3y = 1 \)
\( x + 2y = 11 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x  3y = 1 \)
\( x + 2y = 11 \)
Final Answer 

\( (5, 3) \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x  3y = 1 \)
\( x + 2y = 11 \)
Solution 

Final Answer 

\( (5, 3) \) 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x = 2y + 5 \)
\( 4y = 6x  8 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x = 2y + 5 \)
\( 4y = 6x  8 \)
Final Answer 

no solution
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x = 2y + 5 \)
\( 4y = 6x  8 \)
Solution 

Final Answer 

no solution 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x + 2y = 3 \)
\( 3x  6y = 9 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x + 2y = 3 \)
\( 3x  6y = 9 \)
Final Answer 

infinite number of solutions
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x + 2y = 3 \)
\( 3x  6y = 9 \)
Solution 

Final Answer 

infinite number of solutions 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x + 7y = 5 \)
\( 2x + y = 9 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x + 7y = 5 \)
\( 2x + y = 9 \)
Final Answer 

\( (4, 1) \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x + 7y = 5 \)
\( 2x + y = 9 \)
Solution 

Final Answer 

\( (4, 1) \) 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + 3y = 4 \)
\( x + y = 7 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + 3y = 4 \)
\( x + y = 7 \)
Final Answer 

\( (5, 2) \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + 3y = 4 \)
\( x + y = 7 \)
Solution 

Final Answer 

\( (5, 2) \) 
close solution

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Now you are ready for some 3x3 systems.
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x+3y+z=2; \) \( x+2y+3z=1; \) \( 3x3y+z=0 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x+3y+z=2; \) \( x+2y+3z=1; \) \( 3x3y+z=0 \)
Final Answer 

\( x=4, y=3, z=3 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x+3y+z=2; \) \( x+2y+3z=1; \) \( 3x3y+z=0 \)
Solution 

video by MIP4U 

Final Answer 

\( x=4, y=3, z=3 \) 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2xyz=2; \) \( 4x+yz=5; \) \( 6x2y+2z=17 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2xyz=2; \) \( 4x+yz=5; \) \( 6x2y+2z=17 \)
Final Answer 

\( x=1/2, y=4, z=3 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2xyz=2; \) \( 4x+yz=5; \) \( 6x2y+2z=17 \)
Solution 

video by MIP4U 

Final Answer 

\( x=1/2, y=4, z=3 \) 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x+2z=9; \) \( 2y+z=8; \) \( 4x3y=2 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x+2z=9; \) \( 2y+z=8; \) \( 4x3y=2 \)
Final Answer 

\( x=1, y=2, z=4 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x+2z=9; \) \( 2y+z=8; \) \( 4x3y=2 \)
Solution 

video by PatrickJMT 

Final Answer 

\( x=1, y=2, z=4 \) 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( xy+z=4; \) \( 2x+y+z=7; \) \( x2y+2z=1 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( xy+z=4; \) \( 2x+y+z=7; \) \( x2y+2z=1 \)
Final Answer 

\( x=3, y=0, z=1 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( xy+z=4; \) \( 2x+y+z=7; \) \( x2y+2z=1 \)
Solution 

video by PatrickJMT 

Final Answer 

\( x=3, y=0, z=1 \) 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x+2y3z=1; \) \( 2x+z=0; \) \( 3x4y+4z=2 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x+2y3z=1; \) \( 2x+z=0; \) \( 3x4y+4z=2 \)
Final Answer 

\( x=4/5, y=3/2, z=8/5 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x+2y3z=1; \) \( 2x+z=0; \) \( 3x4y+4z=2 \)
Solution 

video by mattemath 

Final Answer 

\( x=4/5, y=3/2, z=8/5 \) 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x  3y + 3z = 6 \)
\( 2x + 4y + z = 3 \)
\( 3x  5y +4z = 9 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x  3y + 3z = 6 \)
\( 2x + 4y + z = 3 \)
\( 3x  5y +4z = 9 \)
Final Answer 

\( (0, 1, 1) \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x  3y + 3z = 6 \)
\( 2x + 4y + z = 3 \)
\( 3x  5y +4z = 9 \)
Solution 

This problem is solved in 2 consecutive videos.
Final Answer 

\( (0, 1, 1) \) 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x  2y + z = 2 \)
\( 4x + 3y  2z = 4 \)
\( 5x  3y + 3z = 8 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x  2y + z = 2 \)
\( 4x + 3y  2z = 4 \)
\( 5x  3y + 3z = 8 \)
Final Answer 

\( (1, 2, 3) \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x  2y + z = 2 \)
\( 4x + 3y  2z = 4 \)
\( 5x  3y + 3z = 8 \)
Solution 

Final Answer 

\( (1, 2, 3) \) 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x + y  z = 2 \)
\( 2x  y + z = 0 \)
\( x  2y + 3z = 1 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x + y  z = 2 \)
\( 2x  y + z = 0 \)
\( x  2y + 3z = 1 \)
Final Answer 

\( (2/3, 7/3, 1) \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x + y  z = 2 \)
\( 2x  y + z = 0 \)
\( x  2y + 3z = 1 \)
Solution 

Final Answer 

\( (2/3, 7/3, 1) \) 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + y  z = 1 \)
\( 3x + 2y + 2z = 13 \)
\( 4x  2y + 3z = 9 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + y  z = 1 \)
\( 3x + 2y + 2z = 13 \)
\( 4x  2y + 3z = 9 \)
Final Answer 

\( (1, 2, 3) \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + y  z = 1 \)
\( 3x + 2y + 2z = 13 \)
\( 4x  2y + 3z = 9 \)
Solution 

Final Answer 

\( (1, 2, 3) \) 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + 3y  5z = 1 \)
\( x + y  z = 2 \)
\( 2y + z = 8 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + 3y  5z = 1 \)
\( x + y  z = 2 \)
\( 2y + z = 8 \)
Final Answer 

\( (1, 3, 2) \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + 3y  5z = 1 \)
\( x + y  z = 2 \)
\( 2y + z = 8 \)
Solution 

Final Answer 

\( (1, 3, 2) \) 
close solution

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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x + y  z = 0 \)
\( 2x + 5y + 4z = 1 \)
\( 3x + 2y + z = 1 \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x + y  z = 0 \)
\( 2x + 5y + 4z = 1 \)
\( 3x + 2y + z = 1 \)
Final Answer 

\( (1/2, 2/3, 5/6) \)
Problem Statement 

Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x + y  z = 0 \)
\( 2x + 5y + 4z = 1 \)
\( 3x + 2y + z = 1 \)
Solution 

Final Answer 

\( (1/2, 2/3, 5/6) \) 
close solution

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Really UNDERSTAND Precalculus
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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Single Variable Calculus 

MultiVariable Calculus 

Differential Equations 

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