Line Equations  

slope 
\(\displaystyle{ m = \frac{y_2  y_1}{x_2  x_1} }\) 
pointslope form 
\(yy_1 = m(xx_1)\) 
slopeintercept form 
\(y=mx+b\) 
general form 
\( Ax + By = C \) 
Before we get down to the details of lines, let's watch this video to remind us of the slope of a line.
video by Prof Leonard 

In order to determine an equation of a line, you need only two pieces of information, the slope and one point on the line. If you are given two points and no slope, use the slope equation
\(\displaystyle{ m = \frac{y_2  y_1}{x_2  x_1} }\) to find the slope and then use one of the points (either one, its doesn't matter, you will get the same answer no matter which point you choose).
Once you have the slope, usually called m, and one point, \((x_1,y_1)\), you can find the equation of the line using two possible equations.
1. The one that is the easiest and requires the least amount of algebra is the pointslope form, \(yy_1 = m(xx_1)\). Notice that this is just the slope equation in a little different form.
video by Prof Leonard 

2. The second, which works too, is to use the slopeintercept form, \(y=mx+b\) and solve for \(b\). This seems to be the preferred method of most students.
For your final answer, most instructors prefer the slopeintercept form \(y=mx+b\). This is the required form for answers on this site. However, you need to check with your instructor to see what they expect. Here is a video on why we need the slopeintercept form.
video by Prof Leonard 

SlopeIntercept Form of a Line 

Adjust the sliders for the slope m and the yintercept b to see the plot and the equation change. 

As we said above, your goal is to get the slope and one point on the line in order to find the equation of the line. Start with these problems to get some practice on this.
Instructions   Unless otherwise instructed, find the equation of the line, in slopeintercept form, with the given slope passing through the given point.
Find the equation of the line, in slopeintercept form, with slope \( 2/3 \) passing through the point \( (4, 6) \).
Problem Statement 

Find the equation of the line, in slopeintercept form, with slope \( 2/3 \) passing through the point \( (4, 6) \).
Final Answer 

\( y = 2x/3 +10/3 \)
Problem Statement 

Find the equation of the line, in slopeintercept form, with slope \( 2/3 \) passing through the point \( (4, 6) \).
Solution 

Final Answer 

\( y = 2x/3 +10/3 \) 
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Find the equation of the line, in slopeintercept form, with slope \( 2/3 \) passing through the point \( (1, 1) \).
Problem Statement 

Find the equation of the line, in slopeintercept form, with slope \( 2/3 \) passing through the point \( (1, 1) \).
Final Answer 

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

Find the equation of the line, in slopeintercept form, with slope \( 2/3 \) passing through the point \( (1, 1) \).
Solution 

We are given the slope \( m = 2/3 \) and a point on the line \( (1, 1) \). 
Let's use the pointslope form of the equation of the line, \( yy_1 = m(xx_1) \) 
\( y(1) = (2/3)( x 1) \) 
\( y + 1 = 2x/3 + 2/3 \) 
\( y = 2x/3 + 2/3  1 \) 
Final Answer: \( y = 2x/3  1/3 \) 
Here is how we would use the slopeintercept form to solve this problem. 
\( y = mx + b \to 1 = (2/3)(1) + b\) 
\( b = 1 + 2/3 = 1/3 \) 
Final Answer: \( y = 2x/3  1/3 \) 
Final Answer 

\( y = 2x/3  1/3 \) 
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Find the equation of the line, in slopeintercept form, with slope \( 2 \) passing through the point \( (1, 6) \).
Problem Statement 

Find the equation of the line, in slopeintercept form, with slope \( 2 \) passing through the point \( (1, 6) \).
Final Answer 

\( y = 2x  4 \)
Problem Statement 

Find the equation of the line, in slopeintercept form, with slope \( 2 \) passing through the point \( (1, 6) \).
Solution 

Final Answer 

\( y = 2x  4 \) 
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Find the equation of the line, in slopeintercept form, with slope \( 3 \) passing through the point \( (2, 3) \).
Problem Statement 

Find the equation of the line, in slopeintercept form, with slope \( 3 \) passing through the point \( (2, 3) \).
Final Answer 

\( y = 3x + 9 \)
Problem Statement 

Find the equation of the line, in slopeintercept form, with slope \( 3 \) passing through the point \( (2, 3) \).
Solution 

We are given the slope \( m = 3 \) and a point on the line at \( (2, 3) \). 
Let's use the pointslope form \( yy_1 = m(x  x_1) \). 
\( y3 = 3( x  (2)) \) 
\( y  3 = 3x + 6 \) 
\( y = 3x + 6 + 3 \) 
Final Answer: \( y = 3x + 9 \) 
If we used the slopeintercept form to solve this problem, we would have this. 
\( y = mx + b \to 3 = 3(2) + b \) 
\( b = 3 + 6 \to b = 9 \) 
Final Answer: \( y = 3x + 9 \) 
Final Answer 

\( y = 3x + 9 \) 
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Even though your goal is to have the slope and one point on the line, you will not usually be given this exact information in order to find the equation of the line. Sometimes you are given two points. From that information you need to determine the slope and then use one of the points to get the equation of the line. These problems will give you practice with this.
Instructions   Unless otherwise instructed, find the equation of the line that passes through the two given points. Give your answer in slopeintercept form.
Find the equation of the line in slopeintercept form that passes through \((3,7)\) and \((5,1)\).
Problem Statement 

Find the equation of the line in slopeintercept form that passes through \((3,7)\) and \((5,1)\).
Final Answer 

\(y=x+4\)
Problem Statement 

Find the equation of the line in slopeintercept form that passes through \((3,7)\) and \((5,1)\).
Solution 

Final Answer 

\(y=x+4\) 
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Find the equation of the line in slopeintercept form that passes through the points \((3,2)\) and \((2,1)\).
Problem Statement 

Find the equation of the line in slopeintercept form that passes through the points \((3,2)\) and \((2,1)\).
Final Answer 

\( y=x+1 \)
Problem Statement 

Find the equation of the line in slopeintercept form that passes through the points \((3,2)\) and \((2,1)\).
Solution 

Final Answer 

\( y=x+1 \) 
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Find the equation of the line in slopeintercept form that contains \( (5,0) \) and \( (4,3) \).
Problem Statement 

Find the equation of the line in slopeintercept form that contains \( (5,0) \) and \( (4,3) \).
Final Answer 

\( y = x/3 + 5/3 \)
Problem Statement 

Find the equation of the line in slopeintercept form that contains \( (5,0) \) and \( (4,3) \).
Solution 

Final Answer 

\( y = x/3 + 5/3 \) 
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Determine the equation of the line passing through the points \( (1, 1) \) and \( (5, 1) \), giving your answer in slopeintercept form.
Problem Statement 

Determine the equation of the line passing through the points \( (1, 1) \) and \( (5, 1) \), giving your answer in slopeintercept form.
Final Answer 

\( y = x/2 + 3/2 \)
Problem Statement 

Determine the equation of the line passing through the points \( (1, 1) \) and \( (5, 1) \), giving your answer in slopeintercept form.
Solution 

Final Answer 

\( y = x/2 + 3/2 \) 
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Determine the equation of the line passing through the points \( (1, 1) \) and \( (1, 7) \), giving your answer in standard form \( Ax + By = C \).
Problem Statement 

Determine the equation of the line passing through the points \( (1, 1) \) and \( (1, 7) \), giving your answer in standard form \( Ax + By = C \).
Final Answer 

\( 3x  y = 4 \)
Problem Statement 

Determine the equation of the line passing through the points \( (1, 1) \) and \( (1, 7) \), giving your answer in standard form \( Ax + By = C \).
Solution 

Final Answer 

\( 3x  y = 4 \) 
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Determine the equation of the line passing through \( (2, 3) \) and \( (4, 2) \). Give your answer in slopeintercept form.
Problem Statement 

Determine the equation of the line passing through \( (2, 3) \) and \( (4, 2) \). Give your answer in slopeintercept form.
Final Answer 

\( y = x/6  8/3 \)
Problem Statement 

Determine the equation of the line passing through \( (2, 3) \) and \( (4, 2) \). Give your answer in slopeintercept form.
Solution 

Final Answer 

\( y = x/6  8/3 \) 
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Find the equation of the line that goes through \( (3, 5) \) and \( (2, 8) \). Use the pointslope form \( y  y_1 = m(x  x_1) \) and the second point. Give your answer in slopeintercept form.
Problem Statement 

Find the equation of the line that goes through \( (3, 5) \) and \( (2, 8) \). Use the pointslope form \( y  y_1 = m(x  x_1) \) and the second point. Give your answer in slopeintercept form.
Final Answer 

\( y = 3x/5+ 34/5 \)
Problem Statement 

Find the equation of the line that goes through \( (3, 5) \) and \( (2, 8) \). Use the pointslope form \( y  y_1 = m(x  x_1) \) and the second point. Give your answer in slopeintercept form.
Solution 

Final Answer 

\( y = 3x/5+ 34/5 \) 
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Determine the equation of the line with an xintercept of \(3\) and a yintercept of \(4\). Give your answer in slopeintercept form.
Problem Statement 

Determine the equation of the line with an xintercept of \(3\) and a yintercept of \(4\). Give your answer in slopeintercept form.
Final Answer 

\( y = 4x/3  4 \)
Problem Statement 

Determine the equation of the line with an xintercept of \(3\) and a yintercept of \(4\). Give your answer in slopeintercept form.
Solution 

Final Answer 

\( y = 4x/3  4 \) 
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Okay, before going on, here is great video on what we are going to discuss next, parallel and perpendicular lines.
Parallel Lines
When two lines are parallel, their slopes are equal. So, for two lines in slope intercept form \(y = mx + b_1\) and \( y=mx + b_2 \), the slopes \(m\) are the same but \( b_1 \neq b_2 \) (unless of course they are the same line). One of the nice things about the slope intercept form, \(y = mx + b\) is that you can pull the slope \(m\) and the yintercept \(b\) directly from the equation without having to do any work. For the two parallel lines, the slopes are the same but they cross the yaxis at different points.
Instructions   Unless otherwise instructed, find the equation of the line in slopeintercept parallel to the given line going through the given point.
Find the equation of the line in slopeintercept parallel to \( 3xy/4 = 2 \) going through the point \((1/6,8)\).
Problem Statement 

Find the equation of the line in slopeintercept parallel to \( 3xy/4 = 2 \) going through the point \((1/6,8)\).
Final Answer 

\(y=12x10\)
Problem Statement 

Find the equation of the line in slopeintercept parallel to \( 3xy/4 = 2 \) going through the point \((1/6,8)\).
Solution 

Final Answer 

\(y=12x10\) 
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Write the equation of the line that is parallel to \( x  2y = 6 \) and passes through the point \( (4, 3) \), giving your answer in slopeintercept form.
Problem Statement 

Write the equation of the line that is parallel to \( x  2y = 6 \) and passes through the point \( (4, 3) \), giving your answer in slopeintercept form.
Final Answer 

\( y = x/2 + 5 \)
Problem Statement 

Write the equation of the line that is parallel to \( x  2y = 6 \) and passes through the point \( (4, 3) \), giving your answer in slopeintercept form.
Solution 

Final Answer 

\( y = x/2 + 5 \) 
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Determine the equation of the line that is parallel to \( y = 5x + 4 \) and passes through \( (1, 4) \), giving your answer in slopeintercept form.
Problem Statement 

Determine the equation of the line that is parallel to \( y = 5x + 4 \) and passes through \( (1, 4) \), giving your answer in slopeintercept form.
Final Answer 

\( y = 5x  1 \)
Problem Statement 

Determine the equation of the line that is parallel to \( y = 5x + 4 \) and passes through \( (1, 4) \), giving your answer in slopeintercept form.
Solution 

Final Answer 

\( y = 5x  1 \) 
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Determine the equation of the line that is parallel to \( 3x+2y = 12 \) and passes through \( (2, 5) \), giving your answer in slopeintercept form.
Problem Statement 

Determine the equation of the line that is parallel to \( 3x+2y = 12 \) and passes through \( (2, 5) \), giving your answer in slopeintercept form.
Final Answer 

\( y = 3x/2 + 6 \)
Problem Statement 

Determine the equation of the line that is parallel to \( 3x+2y = 12 \) and passes through \( (2, 5) \), giving your answer in slopeintercept form.
Solution 

Final Answer 

\( y = 3x/2 + 6 \) 
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Find the equation of the line parallel to the line \( y = 2x + 1 \) that passes through the point \( (1,5) \), giving your answer in slopeintercept form.
Problem Statement 

Find the equation of the line parallel to the line \( y = 2x + 1 \) that passes through the point \( (1,5) \), giving your answer in slopeintercept form.
Final Answer 

\( y = 2x + 3 \)
Problem Statement 

Find the equation of the line parallel to the line \( y = 2x + 1 \) that passes through the point \( (1,5) \), giving your answer in slopeintercept form.
Solution 

Final Answer 

\( y = 2x + 3 \) 
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Find the equation of the line parallel to the line \( y = 3x  4 \) that passes through the point \( (6, 4) \), giving your answer in slopeintercept form.
Problem Statement 

Find the equation of the line parallel to the line \( y = 3x  4 \) that passes through the point \( (6, 4) \), giving your answer in slopeintercept form.
Final Answer 

\( y = 3x  14 \)
Problem Statement 

Find the equation of the line parallel to the line \( y = 3x  4 \) that passes through the point \( (6, 4) \), giving your answer in slopeintercept form.
Solution 

Final Answer 

\( y = 3x  14 \) 
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Find the equation of the line parallel to the line \( y = 3x/2 + 3 \) that passes through the point \( (4, 3) \), giving your answer in slopeintercept form.
Problem Statement 

Find the equation of the line parallel to the line \( y = 3x/2 + 3 \) that passes through the point \( (4, 3) \), giving your answer in slopeintercept form.
Final Answer 

\( y = 3x/2  9 \)
Problem Statement 

Find the equation of the line parallel to the line \( y = 3x/2 + 3 \) that passes through the point \( (4, 3) \), giving your answer in slopeintercept form.
Solution 

Final Answer 

\( y = 3x/2  9 \) 
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Find the equation of the line in slopeintercept parallel to \( x+7y=4 \) through the point \((7,9)\).
Problem Statement 

Find the equation of the line in slopeintercept parallel to \( x+7y=4 \) through the point \((7,9)\).
Final Answer 

\( y=x/7+10\)
Problem Statement 

Find the equation of the line in slopeintercept parallel to \( x+7y=4 \) through the point \((7,9)\).
Solution 

Final Answer 

\( y=x/7+10\) 
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Perpendicular Lines
In the case of perpendicular lines, if the equation of one line is \(y = mx + b_1\), the equation of the other is \(y=x/m+b_2\) So the slope of the second line is said to be the negative reciprocal of the first line.
Okay, time for the remaining practice problems.
Instructions   Unless otherwise instructed, Find the equation of the line, in slopeintercept form, perpendicular to the given line going through the given point.
Find the equation of the line, in slopeintercept form, perpendicular to \( 9x+12y=2 \) containing the point \((15,7)\).
Problem Statement 

Find the equation of the line, in slopeintercept form, perpendicular to \( 9x+12y=2 \) containing the point \((15,7)\).
Final Answer 

\( y=4x/3  13 \)
Problem Statement 

Find the equation of the line, in slopeintercept form, perpendicular to \( 9x+12y=2 \) containing the point \((15,7)\).
Solution 

Final Answer 

\( y=4x/3  13 \) 
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Find the equation of the line, in slopeintercept form, perpendicular to \(10x+2y=9\) going through the point \((20,3)\).
Problem Statement 

Find the equation of the line, in slopeintercept form, perpendicular to \(10x+2y=9\) going through the point \((20,3)\).
Final Answer 

\( y = x/5  1 \)
Problem Statement 

Find the equation of the line, in slopeintercept form, perpendicular to \(10x+2y=9\) going through the point \((20,3)\).
Solution 

Final Answer 

\( y = x/5  1 \) 
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Determine the equation of the line that is perpendicular to \( x  4y = 15 \) and passes through \( (3, 1) \), giving your answer in slopeintercept form.
Problem Statement 

Determine the equation of the line that is perpendicular to \( x  4y = 15 \) and passes through \( (3, 1) \), giving your answer in slopeintercept form.
Final Answer 

\( y = 4x + 13 \)
Problem Statement 

Determine the equation of the line that is perpendicular to \( x  4y = 15 \) and passes through \( (3, 1) \), giving your answer in slopeintercept form.
Solution 

Final Answer 

\( y = 4x + 13 \) 
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Write the equation of the line that contains the point \( (3, 5) \) and is perpendicular to the line \( y = 3x + 2 \), giving your answer in slopeintercept form.
Problem Statement 

Write the equation of the line that contains the point \( (3, 5) \) and is perpendicular to the line \( y = 3x + 2 \), giving your answer in slopeintercept form.
Final Answer 

\( y = x/3 + 4 \)
Problem Statement 

Write the equation of the line that contains the point \( (3, 5) \) and is perpendicular to the line \( y = 3x + 2 \), giving your answer in slopeintercept form.
Solution 

Final Answer 

\( y = x/3 + 4 \) 
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Determine the equation of the line that is perpendicular to \( y = 2x + 1 \) and passes through \( (4, 5) \), giving your answer in slopeintercept form.
Problem Statement 

Determine the equation of the line that is perpendicular to \( y = 2x + 1 \) and passes through \( (4, 5) \), giving your answer in slopeintercept form.
Final Answer 

\( y = x/2 + 3 \)
Problem Statement 

Determine the equation of the line that is perpendicular to \( y = 2x + 1 \) and passes through \( (4, 5) \), giving your answer in slopeintercept form.
Solution 

Final Answer 

\( y = x/2 + 3 \) 
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Find the equation of the line perpendicular to the line \( y = 5x  3 \) that passes through the point \( (2, 1) \), giving your answer in slopeintercept form.
Problem Statement 

Find the equation of the line perpendicular to the line \( y = 5x  3 \) that passes through the point \( (2, 1) \), giving your answer in slopeintercept form.
Final Answer 

\( y = x/5  3/5 \)
Problem Statement 

Find the equation of the line perpendicular to the line \( y = 5x  3 \) that passes through the point \( (2, 1) \), giving your answer in slopeintercept form.
Solution 

Final Answer 

\( y = x/5  3/5 \) 
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Find the equation of the line perpendicular to the line \( y = 3x/4  1 \) that passes through the point \( (8, 3) \), giving your answer in slopeintercept form.
Problem Statement 

Find the equation of the line perpendicular to the line \( y = 3x/4  1 \) that passes through the point \( (8, 3) \), giving your answer in slopeintercept form.
Final Answer 

\( y = 4x/3 + 23/3 \)
Problem Statement 

Find the equation of the line perpendicular to the line \( y = 3x/4  1 \) that passes through the point \( (8, 3) \), giving your answer in slopeintercept form.
Solution 

Final Answer 

\( y = 4x/3 + 23/3 \) 
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Find the equation of the line perpendicular to the line \( y = 2x/3 + 4 \) that passes through the point \( (6, 2) \), giving your answer in slopeintercept form.
Problem Statement 

Find the equation of the line perpendicular to the line \( y = 2x/3 + 4 \) that passes through the point \( (6, 2) \), giving your answer in slopeintercept form.
Final Answer 

\( y = 3x/2  11 \)
Problem Statement 

Find the equation of the line perpendicular to the line \( y = 2x/3 + 4 \) that passes through the point \( (6, 2) \), giving your answer in slopeintercept form.
Solution 

Final Answer 

\( y = 3x/2  11 \) 
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Find the equation of the line in slopeintercept perpendicular to \( y=2x+1 \) through the point \((2,0)\).
Problem Statement 

Find the equation of the line in slopeintercept perpendicular to \( y=2x+1 \) through the point \((2,0)\).
Final Answer 

\( y = x/2 + 1 \)
Problem Statement 

Find the equation of the line in slopeintercept perpendicular to \( y=2x+1 \) through the point \((2,0)\).
Solution 

Final Answer 

\( y = x/2 + 1 \) 
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Find the equation of the line in slopeintercept perpendicular to \( y=9x+5 \) through the point \((3,9)\).
Problem Statement 

Find the equation of the line in slopeintercept perpendicular to \( y=9x+5 \) through the point \((3,9)\).
Final Answer 

\( y = x/9 + 26/3 \)
Problem Statement 

Find the equation of the line in slopeintercept perpendicular to \( y=9x+5 \) through the point \((3,9)\).
Solution 

Final Answer 

\( y = x/9 + 26/3 \) 
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Here is a playlist of the videos on this page.
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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