# Trig Inversesap Calculus

Related Pages
Lessons On Trigonometry
Inverse trigonometry
Trigonometric Derivatives
Calculus: Derivatives
Calculus Lessons

### Table Of Derivatives Of Inverse Trigonometric Functions

The following table gives the formula for the derivatives of the inverse trigonometric functions. Scroll down the page for more examples and solutions on how to use the formulas.

Derivatives of Inverse Trigonometric Functions AP CALCULUS. Calculator Conversion Identities sec cot csc x x x -1 (l/x) cos 7/2 — tan-I x sin-I (l/x). Problems are periodic and trigonometric spectral methods suffer from Gibbs’ phenomenon23 if they are applied to non-periodic problems. This work deals with the system identiﬁcation on non-periodic domains and is a sequel to our previous work.2 Our attempt is to extend the algebraic spectral operators to the. These are the Lecture Notes of Calculus which includes Integration of Rational Functions, Partial Fraction, Cover Up, Tan, Divided By etc. Key important points are: Inverse functions, Derivatives, One to One, Horizontal Line Test, Inverse, Properties. Search this site. CSH AP Calculus.

Example:
Differentiate

Solution:
We can use the above formula and the chain rule.

Example:
Differentiate

Solution:
We use the product rule and chain rule.

Inverse Trigonometric Functions - Derivatives

Formulas for the derivatives of the six inverse trig functions and derivative examples.

Examples:
Find the derivatives of the following functions

1. f(x) = (sin-1)2
2. g(t) = cos-1√(2t - 1)
3. y = tan-1(x/a) + ln√((x-a)/(x+a))
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Inverse Trigonometric Functions - Derivatives - Harder Example

Example:
Find the derivatives of
y = sec-1√(1 + x2)

Inverse Trigonometric Functions - Derivatives - Harder Example

Example:
Find the derivatives of
y = sin-1(cos x/(1+sinx))

• Show Video Lesson

Derivatives of Inverse Trig Functions

One example does not require the chain rule and one example requires the chain rule.

Examples:
Find the derivatives of each given function.

1. f(x) = 3sin-1(x)
2. g(x) = 4cos-1(3x2)

Derivatives of Inverse Trig Functions

Examples:
Find the derivatives of each given function.

1. f(x) = -2cot-1(x)
2. g(x) = 5tan-1(2x)
• Show Video Lesson

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### Section 1-3 : Trig Functions

The intent of this section is to remind you of some of the more important (from a Calculus standpoint…) topics from a trig class. One of the most important (but not the first) of these topics will be how to use the unit circle. We will leave the most important topic to the next section.

First let’s start with the six trig functions and how they relate to each other.

[begin{array}{ll}{cos left( x right)}&hspace{0.75in}sin left( x right){tan left( x right) = displaystyle frac{{sin left( x right)}}{{cos left( x right)}}}&hspace{0.75in}cot left( x right) = displaystyle frac{{cos left( x right)}}{{sin left( x right)}} = displaystyle frac{1}{{tan left( x right)}}{sec left( x right) = displaystyle frac{1}{{cos left( x right)}}}&hspace{0.75in}csc left( x right) = displaystyle frac{1}{{sin left( x right)}}end{array}]

Recall as well that all the trig functions can be defined in terms of a right triangle.

From this right triangle we get the following definitions of the six trig functions.

 (displaystyle cos theta = frac{{{rm{adjacent}}}}{{{rm{hypotenuse}}}}) (displaystyle sin theta = frac{{{rm{opposite}}}}{{{rm{hypotenuse}}}}) (displaystyle tan theta = frac{{{rm{opposite}}}}{{{rm{adjacent}}}}) (displaystyle cot theta = frac{{{rm{adjacent}}}}{{{rm{opposite}}}}) (displaystyle sec theta = frac{{{rm{hypotenuse}}}}{{{rm{adjacent}}}}) (displaystyle csc theta = frac{{{rm{hypotenuse}}}}{{{rm{opposite}}}})

### Trig Inversesap Calculus Worksheet

Remembering both the relationship between all six of the trig functions and their right triangle definitions will be useful in this course on occasion.

Next, we need to touch on radians. In most trig classes instructors tend to concentrate on doing everything in terms of degrees (probably because it’s easier to visualize degrees). The same is true in many science classes. However, in a calculus course almost everything is done in radians. The following table gives some of the basic angles in both degrees and radians.

 Degree 0 30 45 60 90 180 270 360 Radians 0 (displaystyle frac{pi }{6}) (displaystyle frac{pi }{4}) (displaystyle frac{pi }{3}) (displaystyle frac{pi }{2}) (pi ) (displaystyle frac{{3pi }}{2}) (2pi )

Know this table! We may not see these specific angles all that much when we get into the Calculus portion of these notes, but knowing these can help us to visualize each angle. Now, one more time just make sure this is clear.

Be forewarned, everything in most calculus classes will be done in radians!

Let’s next take a look at one of the most overlooked ideas from a trig class. The unit circle is one of the more useful tools to come out of a trig class. Unfortunately, most people don’t learn it as well as they should in their trig class.

Below is unit circle with just the first quadrant filled in with the “standard” angles. The way the unit circle works is to draw a line from the center of the circle outwards corresponding to a given angle. Then look at the coordinates of the point where the line and the circle intersect. The first coordinate, i.e. the (x)-coordinate, is the cosine of that angle and the second coordinate, i.e. the (y)-coordinate, is the sine of that angle. We’ve put some of the basic angles along with the coordinates of their intersections on the unit circle.

So, from the unit circle above we can see that (cos left( {frac{pi }{6}} right) = frac{{sqrt 3 }}{2}) and (sin left( {frac{pi }{6}} right) = frac{1}{2}).

Also, remember how the signs of angles work. If you rotate in a counter clockwise direction the angle is positive and if you rotate in a clockwise direction the angle is negative.

Recall as well that one complete revolution is (2pi ), so the positive (x)-axis can correspond to either an angle of 0 or (2pi ) (or (4pi ), or (6pi ), or ( - 2pi ), or ( - 4pi ), etc. depending on the direction of rotation). Likewise, the angle ({textstyle{pi over 6}}) (to pick an angle completely at random) can also be any of the following angles:

(displaystyle frac{pi }{6} + 2pi = frac{{13pi }}{6}) (start at (displaystyle frac{pi }{6}) then rotate once around counter clockwise)

(displaystyle frac{pi }{6} + 4pi = frac{{25pi }}{6}) (start at (displaystyle frac{pi }{6}) then rotate around twice counter clockwise)

(displaystyle frac{pi }{6} - 2pi = - frac{{11pi }}{6}) (start at (displaystyle frac{pi }{6}) then rotate once around clockwise)

(displaystyle frac{pi }{6} - 4pi = - frac{{23pi }}{6}) (start at (displaystyle frac{pi }{6}) then rotate around twice clockwise)

etc.

In fact, ({textstyle{pi over 6}}) can be any of the following angles ({frac{pi}{6}} + 2pi ,n,;;n = 0, pm 1, pm 2, pm 3, ldots ) In this case (n) is the number of complete revolutions you make around the unit circle starting at ({frac{pi}{6}}). Positive values of (n) correspond to counter clockwise rotations and negative values of (n) correspond to clockwise rotations.

So, why did we only put in the first quadrant? The answer is simple. If you know the first quadrant then you can get all the other quadrants from the first with a small application of geometry. You’ll see how this is done in the following set of examples.

Example 1 Evaluate each of the following.
1. (sin left( {frac{{2pi }}{3}} right)) and (sin left( { - frac{{2pi }}{3}} right))
2. (cos left( {frac{{7pi }}{6}} right)) and (cos left( { - frac{{7pi }}{6}} right))
3. (tan left( { - frac{pi }{4}} right)) and (tan left( {frac{{7pi }}{4}} right))
4. (sec left( {frac{{25pi }}{6}} right))
Show All SolutionsHide All Solutionsa (sin left( {frac{{2pi }}{3}} right)) and (sin left( { - frac{{2pi }}{3}} right)) Show Solution

### Trig Inversesap Calculus Problems

The first evaluation in this part uses the angle (frac{{2pi }}{3}). That’s not on our unit circle above, however notice that (frac{{2pi }}{3} = pi - frac{pi }{3}). So (frac{{2pi }}{3}) is found by rotating up (frac{pi }{3}) from the negative (x)-axis. This means that the line for (frac{{2pi }}{3}) will be a mirror image of the line for (frac{pi }{3}) only in the second quadrant. The coordinates for (frac{{2pi }}{3}) will be the coordinates for (frac{pi }{3}) except the (x) coordinate will be negative.

Likewise, for ( - frac{{2pi }}{3})we can notice that ( - frac{{2pi }}{3} = - pi + frac{pi }{3}), so this angle can be found by rotating down (frac{pi }{3}) from the negative (x)-axis. This means that the line for ( - frac{{2pi }}{3}) will be a mirror image of the line for (frac{pi }{3}) only in the third quadrant and the coordinates will be the same as the coordinates for (frac{pi }{3}) except both will be negative.

Both of these angles, along with the coordinates of the intersection points, are shown on the following unit circle.

From this unit circle we can see that (sin left( {frac{{2pi }}{3}} right) = frac{{sqrt 3 }}{2})and (sin left( { - frac{{2pi }}{3}} right) = - frac{{sqrt 3 }}{2}).

This leads to a nice fact about the sine function. The sine function is called an odd function and so for ANY angle we have

[sin left( { - theta } right) = - sin left( theta right)]
b (cos left( {frac{{7pi }}{6}} right)) and (cos left( { - frac{{7pi }}{6}} right)) Show Solution

For this example, notice that (frac{{7pi }}{6} = pi + frac{pi }{6}) so this means we would rotate down (frac{pi }{6}) from the negative (x)-axis to get to this angle. Also ( - frac{{7pi }}{6} = - pi - frac{pi }{6}) so this means we would rotate up (frac{pi }{6}) from the negative (x)-axis to get to this angle. So, as with the last part, both of these angles will be mirror images of (frac{pi }{6}) in the third and second quadrants respectively and we can use this to determine the coordinates for both of these new angles.

Both of these angles are shown on the following unit circle along with the coordinates for the intersection points.

From this unit circle we can see that (cos left( {frac{{7pi }}{6}} right) = - frac{{sqrt 3 }}{2})and (cos left( { - frac{{7pi }}{6}} right) = - frac{{sqrt 3 }}{2}). In this case the cosine function is called an even function and so for ANY angle we have

[cos left( { - theta } right) = cos left( theta right)]
c (tan left( { - frac{pi }{4}} right)) and (tan left( {frac{{7pi }}{4}} right)) Show Solution

Here we should note that (frac{{7pi }}{4} = 2pi - frac{pi }{4}) so (frac{{7pi }}{4}) and ( - frac{pi }{4}) are in fact the same angle! Also note that this angle will be the mirror image of (frac{pi }{4}) in the fourth quadrant. The unit circle for this angle is

Now, if we remember that (tan left( x right) = frac{{sin left( x right)}}{{cos left( x right)}}) we can use the unit circle to find the values of the tangent function. So,

[tan left( frac{7pi }{4} right)=tan left( -frac{pi }{4} right)=frac{sin left( -{pi }/{4}; right)}{cos left( -{pi }/{4}; right)}=frac{-{sqrt{2}}/{2};}{{sqrt{2}}/{2};}=-1]

On a side note, notice that (tan left( {frac{pi }{4}} right) = 1) and we can see that the tangent function is also called an odd function and so for ANY angle we will have

[tan left( { - theta } right) = - tan left( theta right)]
d (sec left( {frac{{25pi }}{6}} right)) Show Solution

Here we need to notice that (frac{{25pi }}{6} = 4pi + frac{pi }{6}). In other words, we’ve started at (frac{pi }{6}) and rotated around twice to end back up at the same point on the unit circle. This means that

[sec left( {frac{{25pi }}{6}} right) = sec left( {4pi + frac{pi }{6}} right) = sec left( {frac{pi }{6}} right)]

### Trig Inversesap Calculus Calculator

Now, let’s also not get excited about the secant here. Just recall that

[sec left( x right) = frac{1}{{cos left( x right)}}]

and so all we need to do here is evaluate a cosine! Therefore,

[sec left( frac{25pi }{6} right)=sec left( frac{pi }{6} right)=frac{1}{cos left( frac{pi }{6} right)}=frac{1}{{}^{sqrt{3}}/{}_{2}}=frac{2}{sqrt{3}}]

So, in the last example we saw how the unit circle can be used to determine the value of the trig functions at any of the “common” angles. It’s important to notice that all of these examples used the fact that if you know the first quadrant of the unit circle and can relate all the other angles to “mirror images” of one of the first quadrant angles you don’t really need to know whole unit circle. If you’d like to see a complete unit circle I’ve got one on my Trig Cheat Sheet that is available at http://tutorial.math.lamar.edu.

### Trig Inversesap Calculus Formula

Another important idea from the last example is that when it comes to evaluating trig functions all that you really need to know is how to evaluate sine and cosine. The other four trig functions are defined in terms of these two so if you know how to evaluate sine and cosine you can also evaluate the remaining four trig functions.

We’ve not covered many of the topics from a trig class in this section, but we did cover some of the more important ones from a calculus standpoint. There are many important trig formulas that you will use occasionally in a calculus class. Most notably are the half-angle and double-angle formulas. If you need reminded of what these are, you might want to download my Trig Cheat Sheet as most of the important facts and formulas from a trig class are listed there.