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It summarizes function equivalences, cancellation properties, and key domain and range rules for inverse trig functions such as arcsin, arccos, arctan, and arccot. The sheet also collects angle and unit-circle relationships, right-triangle trig links, and foundational theorems/identities including the Pythagorean theorem and standard trigonometric identities used for problem solving.",{"@graph":63,"@context":118},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":46,"@type":70,"position":76},"https://docshare.wps.com/template/paper-templates/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/precalculus-trig-and-inverse-trig-functions-formula-reference-sheet/572139/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/precalculus-trig-and-inverse-trig-functions-formula-reference-sheet/572139.png","ImageObject",442,249,{"name":88,"@type":89},"Supernature","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-15","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":9},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What does the sheet cover for inverse trigonometric functions?","Question",{"text":108,"@type":109},"It defines inverse trig functions and lists their cancellation properties. It also provides domain and range intervals for arcsin, arccos, arctan, and arccot.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"Which identity is included for right-triangle relationships?",{"text":113,"@type":109},"It includes the Pythagorean theorem and the standard right-triangle trigonometry relationships connecting sin, cos, and tan to triangle sides.",{"name":115,"@type":106,"acceptedAnswer":116},"How are even and odd trigonometric identities presented?",{"text":117,"@type":109},"The sheet groups trig parity rules showing which functions are even or odd and how signs change for inputs like -θ.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},572139,1792052527,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":45,"category_name":46,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":9,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":73,"language":135,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":61,"update_tm":139,"read_time":9},962093270004,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","| PRECALCULUS TRIG AND INVERSE TRIG FUNCTIONS |  |  |  |  |\n| --- | --- | --- | --- | --- |\n|  |  |  |  |  |\n| 􀢟 = 􀜛􀜑􀜖 􀢞 |  | 􀢟 = 􀜋􀜗􀜛 􀢞 |  | 􀢟 = 􀜜􀜉􀜖 􀢞 |\n| \u003Cbr>􀛲􀜗􀜕􀜉􀜑􀜖: (−∞, ∞) or − ∞ \u003C 􀝔 \u003C ∞ |  | \u003Cbr>􀛲􀜗􀜕􀜉􀜑􀜖: (−∞, ∞) or − ∞ \u003C 􀝔 \u003C ∞ |  | \u003Cbr>􀛲􀜗􀜕􀜉􀜑􀜖: {􀝔 | 􀝔 ≠ 􀰗2 + 􀝇􀟨, where 􀝇 is an integer } |\n|  |  |  |  |  |\n| 􀜀􀜉􀜖􀜏􀜍: [ −1, 1] or − 1 ≤ sin 􀝔 ≤ 1 |  | 􀜀􀜉􀜖􀜏􀜍: [ −1, 1] or − 1 ≤ sin 􀝔 ≤ 1 |  | 􀜀􀜉􀜖􀜏􀜍: (−∞, ∞) |\n|  |  |  |  |  |\n| 􀛾􀜍􀜚􀜑􀜗􀜌: 2􀟨 |  | 􀛾􀜍􀜚􀜑􀜗􀜌: 2􀟨 |  | 􀜄􀜍􀜚􀜜􀜑􀜋􀜉􀜔 􀛯􀜛􀜡􀜕􀜘􀜜􀜗􀜜􀜍􀜛: 􀝔 =\u003Cbr>􀟨\u003Cbr>2\u003Cbr>+ 􀝇􀟨, 􀝇 an integer |\n| 􀜈􀜍􀜚􀜗􀜛: (􀝇􀟨, 0), where 􀝇 is an integer |  | 􀜈􀜍􀜚􀜗􀜛: (􀝇􀟨2 , 0) , where 􀝇 is an odd integer |  |  |\n|  |  |  |  | 􀛾􀜍􀜚􀜑􀜗􀜌: 􀟨 |\n| 􀜁􀜡􀜕􀜕􀜍􀜜􀜚􀜡: Odd Function |  |  |  |  |\n|  |  | 􀜁􀜡􀜕􀜕􀜍􀜜􀜚􀜡: Even Function |  | 􀜈􀜍􀜚􀜗􀜛: (􀝇􀟨, 0), where 􀝇 is an integer |\n| 􀛵􀜍􀜖􀜍􀜚􀜉􀜔 􀛴􀜗􀜚􀜕: 􀝕 = 􀜽 sin(􀜾􀝔 − 􀜿) + 􀝀 |  |  |  |  |\n|  |  | 􀛵􀜍􀜖􀜍􀜚􀜉􀜔 􀛴􀜗􀜚􀜕: 􀝕 = 􀜽 cos(􀜾􀝔 − 􀜿) + 􀝀 |  | 􀜁􀜡􀜕􀜕􀜍􀜜􀜚􀜡: Odd Function |\n| 􀛯􀜕􀜘􀜔􀜑􀜜􀜝􀜌􀜍: |􀜽|, 􀛾􀜍􀜚􀜑􀜗􀜌:  2􀟨 􀛾􀜐􀜉􀜛􀜍 􀜁􀜐􀜑􀜎􀜜:  􀜿\u003Cbr>|􀜾| , 􀜾 |  |  |  |  |\n|  |  | 􀛯􀜕􀜘􀜔􀜑􀜜􀜝􀜌􀜍: |􀜽|, 􀛾􀜍􀜚􀜑􀜗􀜌:  2􀟨 􀛾􀜐􀜉􀜛􀜍 􀜁􀜐􀜑􀜎􀜜:  􀜿\u003Cbr>|􀜾| , 􀜾 |  | 􀛵􀜍􀜖􀜍􀜚􀜉􀜔 􀛴􀜗􀜚􀜕: 􀝕 = 􀜽 tan(􀜾􀝔 − 􀜿) + 􀝀 |\n|  |  |  |  | 􀛯􀜕􀜘􀜔􀜑􀜜􀜝􀜌􀜍: |􀜽|, 􀛾􀜍􀜚􀜑􀜗􀜌:  􀟨  􀛾􀜐􀜉􀜛􀜍 􀜁􀜐􀜑􀜎􀜜:  􀜿\u003Cbr>|􀜾| , 􀜾 |\n|  |  |  |  |  |\n|  |  |  |  |  |\n\n\n| 􀢟 = 􀜋􀜛􀜋 􀢞 |  | 􀢟 = 􀜛􀜍􀜋 􀢞 |  |  |  | 􀢟 = 􀜋􀜗􀜜 􀢞 |\n| --- | --- | --- | --- | --- | --- | --- |\n| \u003Cbr>􀛲􀜗􀜕􀜉􀜑􀜖: {􀝔| 􀝔 ≠ 􀝇􀟨, where 􀝇 is an integer } |  | \u003Cbr>􀛲􀜗􀜕􀜉􀜑􀜖: {􀝔 | 􀝔 ≠ 􀰗2 + 􀝇􀟨, where 􀝇 is an integer } |  |  |  | \u003Cbr>􀛲􀜗􀜕􀜉􀜑􀜖: {􀝔| 􀝔 ≠ 􀝇􀟨, where 􀝇 is an integer } |\n|  |  |  |  |  |  | 􀜀􀜉􀜖􀜏􀜍: (−∞, ∞) or − ∞ \u003C cot 􀝔 \u003C ∞ |\n| 􀜀􀜉􀜖􀜏􀜍: (−∞, −1] ∪ [1, ∞) |  |  |  |  |  |  |\n|  |  | 􀜀􀜉􀜖􀜏􀜍: (−∞, −1] ∪ [1, ∞) |  |  |  | 􀜄􀜍􀜚􀜜􀜑􀜋􀜉􀜔 􀛯􀜛􀜡􀜕􀜘􀜜􀜗􀜜􀜍􀜛: 􀝔 = 􀝇􀟨, 􀝇 an integer |\n| 􀜄􀜍􀜚􀜜􀜑􀜋􀜉􀜔 􀛯􀜛􀜡􀜕􀜘􀜜􀜗􀜜􀜍􀜛: 􀝔 = 􀝇􀟨, 􀝇 an integer |  |  |  |  |  |  |\n|  |  | 􀜄􀜍􀜚􀜜􀜑􀜋􀜉􀜔 􀛯􀜛􀜡􀜕􀜘􀜜􀜗􀜜􀜍􀜛: 􀝔 = | 􀟨\u003Cbr>2 | + 􀝇􀟨, 􀝇 an integer |  | 􀛾􀜍􀜚􀜑􀜗􀜌: 􀟨 |\n| 􀛾􀜍􀜚􀜑􀜗􀜌: 2􀟨 |  |  |  |  |  |  |\n|  |  |  |  |  |  | 􀜈􀜍􀜚􀜗􀜛: (􀝇􀟨2 , 0) , where 􀝇 is an odd integer |\n|  |  | 􀛾􀜍􀜚􀜑􀜗􀜌: 2􀟨 |  |  |  |  |\n| 􀜈􀜍􀜚􀜗􀜛: None |  |  |  |  |  |  |\n|  |  | 􀜈􀜍􀜚􀜗􀜛: None |  |  |  |  |\n| 􀜁􀜡􀜕􀜕􀜍􀜜􀜚􀜡: Odd Function |  |  |  |  |  | 􀜁􀜡􀜕􀜕􀜍􀜜􀜚􀜡: Odd Function |\n|  |  | 􀜁􀜡􀜕􀜕􀜍􀜜􀜚􀜡: Even Function |  |  |  |  |\n| 􀛵􀜍􀜖􀜍􀜚􀜉􀜔 􀛴􀜗􀜚􀜕: 􀝕 = 􀜽 csc(􀜾􀝔 − 􀜿) + 􀝀 |  |  |  |  |  | 􀛵􀜍􀜖􀜍􀜚􀜉􀜔 􀛴􀜗􀜚􀜕: 􀝕 = 􀜽 cot(􀜾􀝔 − 􀜿) + 􀝀 |\n|  |  | 􀛵􀜍􀜖􀜍􀜚􀜉􀜔 􀛴􀜗􀜚􀜕: 􀝕 = 􀜽 sec(􀜾􀝔 − 􀜿) + 􀝀 |  |  |  |  |\n| 􀛯􀜕􀜘􀜔􀜑􀜜􀜝􀜌􀜍: |􀜽|, 􀛾􀜍􀜚􀜑􀜗􀜌:  2􀟨 􀛾􀜐􀜉􀜛􀜍 􀜁􀜐􀜑􀜎􀜜:  􀜿\u003Cbr>|􀜾| , 􀜾 |  |  |  |  |  | 􀛯􀜕􀜘􀜔􀜑􀜜􀜝􀜌􀜍: |􀜽|, 􀛾􀜍􀜚􀜑􀜗􀜌:  􀟨  􀛾􀜐􀜉􀜛􀜍 􀜁􀜐􀜑􀜎􀜜:  􀜿\u003Cbr>|􀜾| , 􀜾 |\n|  |  | 􀛯􀜕􀜘􀜔􀜑􀜜􀜝􀜌􀜍: |􀜽|, 􀛾􀜍􀜚􀜑􀜗􀜌:  2􀟨 􀛾􀜐􀜉􀜛􀜍 􀜁􀜐􀜑􀜎􀜜:  􀜿\u003Cbr>|􀜾| , 􀜾 |  |  |  |  |\n|  |  |  |  |  |  |  |\n\n\n| INVERSE TRIG DEFINITION |  | INVERSE TRIG: CANCELLATION PROPERTIES |  |  |\n| --- | --- | --- | --- | --- |\n| 􀝕 = sin−1 􀝔 is equivalent to sin 􀝕 = 􀝔 |  | sin(sin−1 􀝔) = 􀝔 for − 1 ≤ 􀝔 ≤ 1 | cos(cos−1 􀝔) = 􀝔 for − 1 ≤ 􀝔 ≤ 1 cos −1(cos 􀝔) = 􀝔 for 0 ≤ 􀝔 ≤ 􀟨 | tan(tan−1 􀝔) = 􀝔 for all 􀝔 . |\n| 􀝕 = cos −1 􀝔 is equivalent to cos 􀝕 = 􀝔 |  |  |  |  |\n|  |  | sin−1(sin 􀝔) = 􀝔 for − 􀟨2 ≤ 􀝔 ≤ 􀟨2 |  | tan−1(tan 􀝔) = 􀝔 for − 􀟨2 \u003C 􀝔 \u003C 􀟨2 |\n| 􀝕 = tan−1 􀝔 is equivalent to tan 􀝕 = 􀝔 |  |  |  |  |\n\n\n| INVERSE TRIG FUNCTIONS: DOMAIN AND RANGE |  |  |  | |\n| --- | --- | --- | --- | --- |\n| FUNCTION | DOMAIN | RANGE |  |  |\n| 􀝕 = sin−1 􀝔 = arcsin 􀝔 | −1 ≤ 􀝔 ≤ 1 |  − 􀟨2 ≤ sin−1 􀝔 ≤ 􀟨2  |  |  |\n| 􀝕 = cos −1 􀝔 = arccos 􀝔 | −1 ≤ 􀝔 ≤ 1 | 0 ≤ cos −1 􀝔 ≤ 􀟨 |  |  |\n| 􀝕 = tan−1 􀝔 = arctan 􀝔 | −∞ ≤ 􀝔 ≤ ∞ |  − 􀟨2 \u003C tan−1 􀝔 \u003C 􀟨2  |  |  |\n| 􀝕 = cot −1 􀝔 = arccot 􀝔 | −∞ ≤ 􀝔 ≤ ∞ | 0 \u003C cot −1 􀝔 \u003C 􀟨 |  |  |\n|  |  |  |  |  |\n\nPRECALCULUS TRIGONOMETRY FORMULAS  \n\n| RIGHT TRIANGLE TRIG. |  |  |  | CIRCLE ANGLE TRIG. |  |  |  |  |  | THE UNIT CIRCLE |  |  |  |  |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| |  |  |  | 􀝕\u003Cbr>(􀝔, 􀝕\u003Cbr>\u003Cbr>􀝕\u003Cbr>)\u003Cbr>\u003Cbr>\u003Cbr>sin 􀟠 = cos 􀟠 = tan 􀟠 =\u003Cbr>􀝕􀝎􀝔􀝎􀝕􀝔 |  | \u003Cbr>􀟠\u003Cbr>􀝔\u003Cbr>csc 􀟠 = sec 􀟠 = cot 􀟠 =\u003Cbr>􀝎􀝕􀝎􀝔􀝔􀝕 |  |  |  | (1,0)\u003Cbr>(0, −1)\u003Cbr>(−1,0)\u003Cbr>(0, 1)\u003Cbr>3\u003Cbr>\u003Cbr>\u003Cbr>\u003Cbr>5􀟨 6\u003Cbr>3􀟨 4\u003Cbr>2􀟨 3\u003Cbr>120°\u003Cbr>135° 150°\u003Cbr>􀟨 2\u003Cbr>90°\u003Cbr>􀟨 3\u003Cbr>60°\u003Cbr>􀟨 4\u003Cbr>45° 30°\u003Cbr>􀟨 6\u003Cbr>\u003Cbr>~~ 􀟨 180°  0°, 360° 0, 2􀟨 ~~\u003Cbr>210° 7􀟨 225°\u003Cbr>6 240° 5􀟨\u003Cbr>    4  4􀟨 3\u003Cbr>\u003Cbr>270°\u003Cbr>3􀟨 2\u003Cbr>330°\u003C","cbCaigI3k95aZcwA","https://ap.wps.com/l/cbCaigI3k95aZcwA","pdf",487168,"English","# Precalculus Trig and Inverse Trig Functions\n## Inverse Trig Definition\n## Inverse Trig Cancellation Properties\n## Inverse Trig Functions: Domain and Range\n## Precalculus Trigonometry Formulas\n## Right Triangle Trig\n## Circle Angle Trig\n## The Unit Circle\n## Pythagorean Theorem\n## Pythagorean Identities\n## Quotient and Reciprocal Identities\n## Even/Odd Identities\n## Double Angle and Half Angle Identities","[{\"question\":\"What does the sheet cover for inverse trigonometric functions?\",\"answer\":\"It defines inverse trig functions and lists their cancellation properties. It also provides domain and range intervals for arcsin, arccos, arctan, and arccot.\"},{\"question\":\"Which identity is included for right-triangle relationships?\",\"answer\":\"It includes the Pythagorean theorem and the standard right-triangle trigonometry relationships connecting sin, cos, and tan to triangle sides.\"},{\"question\":\"How are even and odd trigonometric identities presented?\",\"answer\":\"The sheet groups trig parity rules showing which functions are even or odd and how signs change for inputs like -θ.\"}]","Precalculus Trig and Inverse Trig Functions - Formula Reference Sheet | PDF",1791925756]