{"id":7968,"date":"2023-05-09T22:25:22","date_gmt":"2023-05-09T16:55:22","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=7968"},"modified":"2023-05-07T23:14:57","modified_gmt":"2023-05-07T17:44:57","slug":"how-to-convert-sin-inverse-to-cos-tan-sec-cosec-x-cot-inverse","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/how-to-convert-sin-inverse-to-cos-tan-sec-cosec-x-cot-inverse\/","title":{"rendered":"how to convert sin inverse to cos, tan, sec, cosec x, cot inverse"},"content":{"rendered":"\n

We often need to convert inverse of sin to either inverse of cos, tan, sec, cosec , cot .In this post we will see how we can do it easily<\/p>\n\n\n\n

Inverse of sin to inverse of cos<\/h2>\n\n\n\n

Case 1<\/strong>
$sin^{-1} x$ and x > 0<\/p>\n\n\n\n

Now we can write as<\/p>\n\n\n\n

$\\theta=sin^{-1} x$
$sin \\theta =x$<\/p>\n\n\n\n

Now we know that here $\\theta \\in [0,\\pi\/2]$, so it is an acute angle<\/p>\n\n\n\n

Now it can be written as <\/p>\n\n\n\n

$sin \\theta =\\frac {x}{1} = \\frac {perp}{hyp}$<\/p>\n\n\n\n

In Right angle triangle<\/p>\n\n\n\n

\"\"<\/a><\/figure>\n\n\n\n

Now then base becomes
$\\text{Base} = \\sqrt { 1 -x^2}$<\/p>\n\n\n\n

So , $cos \\theta = \\frac {base}{hyp} = \\sqrt { 1 -x^2}$<\/p>\n\n\n\n

or<\/p>\n\n\n\n

$\\theta= cos^{-1} \\sqrt { 1 -x^2}$
or $sin^{-1} x = cos^{-1} \\sqrt { 1 -x^2}$<\/p>\n\n\n\n

Case <\/strong>II
$sin^{-1} x$ and x < 0<\/p>\n\n\n\n

So value of the function will be in the range $[-\\pi\/2 , 0]$<\/p>\n\n\n\n

Now we know from the property that<\/p>\n\n\n\n

$sin^{-1} (-x)= – sin^{-1} (x)$<\/p>\n\n\n\n

This can be written as
$sin^{-1} x = – sin^{-1} |x| = – cos^{-1} \\sqrt { 1 -|x|^2}= – cos^{-1} \\sqrt { 1 -x^2}$<\/p>\n\n\n\n

This makes sense also as Range of the cos and sin function differ. We can convert with out worrying about the sign in $[0, \\pi\/2] as it is common<\/p>\n\n\n\n

Thus ,we have different formula depending on the values of x<\/p>\n\n\n\n

Inverse of sin to inverse of tan<\/h2>\n\n\n\n

Case 1<\/strong>
$sin^{-1} x$ and x > 0<\/p>\n\n\n\n

from the above, we can write that<\/p>\n\n\n\n

$tan \\theta = \\frac {perp}{base} = \\frac {x}{\\sqrt {1-x^2}}$<\/p>\n\n\n\n

or <\/p>\n\n\n\n

$sin^{-1} x = tan^{-1} \\frac {x}{\\sqrt { 1 -x^2}}$<\/p>\n\n\n\n

Case <\/strong>II
$sin^{-1} x$ and x < 0<\/p>\n\n\n\n

Now we know from the property that<\/p>\n\n\n\n

$sin^{-1} (-x)= – sin^{-1} (x)$<\/p>\n\n\n\n

This can be written as
$sin^{-1} x = – sin^{-1} |x| = – tan^{-1} \\frac {x}{\\sqrt { 1 -x^2}}= tan^{-1} \\frac {-|x|}{\\sqrt { 1 -x^2}} = tan^{-1} \\frac {x}{\\sqrt { 1 -x^2}}$<\/p>\n\n\n\n

So we have same formula for any values of x<\/p>\n\n\n\n

$sin^{-1} x = tan^{-1} \\frac {x}{\\sqrt { 1 -x^2}}$<\/p>\n\n\n\n

Inverse of sin to inverse of cosec<\/h2>\n\n\n\n

This we already know from the property<\/p>\n\n\n\n

$sin^{-1} x = cosec^{-1} \\frac {1}{x}$ <\/p>\n\n\n\n

for all values of x [-1,1]<\/p>\n\n\n\n

Inverse of sin to inverse of sec<\/h2>\n\n\n\n

Case 1<\/strong>
$sin^{-1} x$ and x > 0<\/p>\n\n\n\n

from the above, we can write that<\/p>\n\n\n\n

$sec \\theta = \\frac {hyp}{base} = \\frac {1}{\\sqrt {1-x^2}}$<\/p>\n\n\n\n

or <\/p>\n\n\n\n

$sin^{-1} x = sec^{-1} \\frac {1}{\\sqrt { 1 -x^2}}$<\/p>\n\n\n\n

Case <\/strong>II
$sin^{-1} x$ and x < 0<\/p>\n\n\n\n

Now we know from the property that<\/p>\n\n\n\n

$sin^{-1} (-x)= – sin^{-1} (x)$<\/p>\n\n\n\n

This can be written as
$sin^{-1} x = – sin^{-1} |x| = – sec^{-1} \\frac {1}{\\sqrt { 1 -x^2}} = – sec^{-1} \\frac {1}{\\sqrt { 1 -x^2}}$<\/p>\n\n\n\n

This makes sense also as Range of the sec and sin function differ. We can convert with out worrying about the sign in $[0, \\pi\/2] as it is common<\/p>\n\n\n\n

Thus ,we have different formula depending on the values of x<\/p>\n\n\n\n

Inverse of sin to inverse of cot<\/h2>\n\n\n\n

Case 1<\/strong>
$sin^{-1} x$ and x > 0<\/p>\n\n\n\n

from the above, we can write that<\/p>\n\n\n\n

$cot \\theta = \\frac {base}{perp} = \\frac{\\sqrt {1-x^2}}{x}$<\/p>\n\n\n\n

or <\/p>\n\n\n\n

$sin^{-1} x = cot^{-1} \\frac {\\sqrt { 1 -x^2}}{x}$<\/p>\n\n\n\n

Case <\/strong>II
$sin^{-1} x$ and x < 0<\/p>\n\n\n\n

Now we know from the property that<\/p>\n\n\n\n

$sin^{-1} (-x)= – sin^{-1} (x)$<\/p>\n\n\n\n

This can be written as
$sin^{-1} x = – sin^{-1} |x| = – cot^{-1} \\frac {\\sqrt { 1 -x^2}}{|x|}$<\/p>\n\n\n\n

This makes sense also as Range of the sec and sin function differ. We can convert with out worrying about the sign in $[0, \\pi\/2] as it is common<\/p>\n\n\n\n

Thus ,we have different formula depending on the values of x<\/p>\n","protected":false},"excerpt":{"rendered":"

We often need to convert inverse of sin to either inverse of cos, tan, sec, cosec , cot .In this post we will see how we can do it easily Inverse of sin to inverse of cos Case 1$sin^{-1} x$ and x > 0 Now we can write as $\\theta=sin^{-1} x$$sin \\theta =x$ Now we […]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center 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often need to convert inverse of sin to either inverse of cos, tan, sec, cosec , cot .In this post we will see how we can do it easily Inverse of sin to inverse of cos Case 1$sin^{-1} x$ and x > 0 Now we can write as $\\theta=sin^{-1} x$$sin \\theta =x$ Now we…","_links":{"self":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/7968","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/users\/8"}],"replies":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/comments?post=7968"}],"version-history":[{"count":1,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/7968\/revisions"}],"predecessor-version":[{"id":7970,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/7968\/revisions\/7970"}],"wp:attachment":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/media?parent=7968"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/categories?post=7968"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/tags?post=7968"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}