Difference between revisions of "The inverse sine, cosine and tangent functions"
Jump to navigation
Jump to search
(3 intermediate revisions by one other user not shown) | |||
Line 13: | Line 13: | ||
Example: <math> \sin{\frac{\pi}{4}} = \frac{\sqrt{2}}{2}</math>, so <math>\arcsin{\frac{\sqrt{2}}{2}} = \frac{\pi}{4} </math>. Even though <math> \sin{\frac{3\pi}{4}} = \frac{\sqrt{2}}{2}</math> as well, <math> \frac{3\pi}{4} </math> is outside of the range for <math> \arcsin{x} </math>. | Example: <math> \sin{\frac{\pi}{4}} = \frac{\sqrt{2}}{2}</math>, so <math>\arcsin{\frac{\sqrt{2}}{2}} = \frac{\pi}{4} </math>. Even though <math> \sin{\frac{3\pi}{4}} = \frac{\sqrt{2}}{2}</math> as well, <math> \frac{3\pi}{4} </math> is outside of the range for <math> \arcsin{x} </math>. | ||
+ | |||
+ | Notes: <math>\csc{x} = 1/\sin{x}</math>, and <math>\arccsc{x} = \arcsin{(1/x)}</math>. <math>\arccsc{x}</math> does NOT equal <math> 1/\arcsin{x} </math>. Similarly, <math>\arcsec{x} = \arccos{(1/x)}</math>, and NOT <math> 1/\arccos{x} </math>. | ||
+ | |||
+ | ===Inverse Trig Functions=== | ||
+ | |||
+ | <math>\int \arcsin u \mathrm{d}u = u \arcsin u + \sqrt {1 - u^2} + C</math> | ||
+ | |||
+ | <p><math>\int \arccos u \mathrm{d}u = u \arccos u - \sqrt {1 - u^2} + C</math></p> | ||
+ | |||
+ | <p><math>\int \arctan u \mathrm{d}u = u \arctan u - \ln \sqrt {1 + u^2} + C</math></p> | ||
+ | |||
+ | <p><math>\int \arccot u \mathrm{d}u = u \arccot u + \ln \sqrt {1 + u^2} + C</math></p> | ||
+ | |||
+ | <p><math>\int \arcsec u \mathrm{d}u = u \arcsec u + \ln \left \vert u + \sqrt {u^2 -1} \right\vert + C</math></p> | ||
+ | |||
+ | <p><math>\int \arccsc u \mathrm{d}u = u \arccsc u + \ln \left \vert u + \sqrt {u^2 - 1} \right \vert + C</math></p> | ||
+ | |||
==Resources== | ==Resources== | ||
* [https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-solve-for-an-angle/a/inverse-trig-functions-intro Intro to Inverse Trig Functions], Khan Academy | * [https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-solve-for-an-angle/a/inverse-trig-functions-intro Intro to Inverse Trig Functions], Khan Academy | ||
* [https://tutorial.math.lamar.edu/extras/algebratrigreview/inversetrig.aspx Inverse Trig Functions], Paul's Online Notes | * [https://tutorial.math.lamar.edu/extras/algebratrigreview/inversetrig.aspx Inverse Trig Functions], Paul's Online Notes |
Latest revision as of 22:29, 9 October 2021
Notation of Inverse Trig Functions
. The domain of is , and its range is .
. Domain: ; range: .
. Domain: ; range: .
. Domain: ; range: .
. Domain: ; range: .
. Domain: ; range: .
Example: , so . Even though as well, is outside of the range for .
Notes: , and . does NOT equal . Similarly, , and NOT .
Inverse Trig Functions
Resources
- Intro to Inverse Trig Functions, Khan Academy
- Inverse Trig Functions, Paul's Online Notes