{"id":"https://openalex.org/W229829915","doi":"https://doi.org/10.5951/mt.65.5.0431","title":"Visual Aids for Relating Direct and Inverse Circular Functions","display_name":"Visual Aids for Relating Direct and Inverse Circular Functions","publication_year":1972,"publication_date":"1972-05-01","ids":{"openalex":"https://openalex.org/W229829915","doi":"https://doi.org/10.5951/mt.65.5.0431","mag":"229829915"},"language":"en","primary_location":{"id":"doi:10.5951/mt.65.5.0431","doi":"https://doi.org/10.5951/mt.65.5.0431","is_oa":false,"landing_page_url":"https://doi.org/10.5951/mt.65.5.0431","pdf_url":null,"source":{"id":"https://openalex.org/S2764739392","display_name":"Mathematics Teacher Learning and Teaching PK-12","issn_l":"0025-5769","issn":["0025-5769","2330-0582"],"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/P4310315778","host_organization_name":"National Council of Teachers of Mathematics","host_organization_lineage":["https://openalex.org/P4310315778"],"host_organization_lineage_names":["National Council of Teachers of Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Mathematics Teacher","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5016698392","display_name":"William K. Viertel","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I159428640","display_name":"SUNY Canton","ror":"https://ror.org/00tdmgj61","country_code":"US","type":"education","lineage":["https://openalex.org/I159428640"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"William K. Viertel","raw_affiliation_strings":["1 State University Agricultural and Technical College Canton, New York"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"1 State University Agricultural and Technical College Canton, New York","institution_ids":["https://openalex.org/I159428640"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5016698392"],"corresponding_institution_ids":["https://openalex.org/I159428640"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.07555959,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"65","issue":"5","first_page":"431","last_page":"431"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10130","display_name":"Mathematics Education and Teaching Techniques","score":0.9678000211715698,"subfield":{"id":"https://openalex.org/subfields/3304","display_name":"Education"},"field":{"id":"https://openalex.org/fields/33","display_name":"Social Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T10130","display_name":"Mathematics Education and Teaching Techniques","score":0.9678000211715698,"subfield":{"id":"https://openalex.org/subfields/3304","display_name":"Education"},"field":{"id":"https://openalex.org/fields/33","display_name":"Social Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T12504","display_name":"Mathematics and Applications","score":0.028699999675154686,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T13806","display_name":"Mathematical and Computational Methods","score":0.0010000000474974513,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/inverse-trigonometric-functions","display_name":"inverse trigonometric functions","score":0.9319999814033508},{"id":"https://openalex.org/keywords/visual-aids","display_name":"visual aids","score":0.8069999814033508},{"id":"https://openalex.org/keywords/one-to-one-functions","display_name":"one-to-one functions","score":0.7839999794960022},{"id":"https://openalex.org/keywords/domain-and-range","display_name":"domain and range","score":0.8100000023841858},{"id":"https://openalex.org/keywords/trigonometry-instruction","display_name":"trigonometry instruction","score":0.7789999842643738}],"concepts":[{"id":"https://openalex.org/C178009071","wikidata":"https://www.wikidata.org/wiki/Q93344","display_name":"Trigonometric functions","level":2,"score":0.8385152816772461},{"id":"https://openalex.org/C140528856","wikidata":"https://www.wikidata.org/wiki/Q191884","display_name":"Inverse function","level":3,"score":0.7409995794296265},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.7196594476699829},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.6783267259597778},{"id":"https://openalex.org/C29001434","wikidata":"https://www.wikidata.org/wiki/Q8084","display_name":"Trigonometry","level":2,"score":0.6197553873062134},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4913569986820221},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.46733129024505615},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.45866456627845764},{"id":"https://openalex.org/C204323151","wikidata":"https://www.wikidata.org/wiki/Q905424","display_name":"Range (aeronautics)","level":2,"score":0.42772233486175537},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4009053409099579},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.33416515588760376},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.23249968886375427},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.1063385009765625},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.10021331906318665},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C146978453","wikidata":"https://www.wikidata.org/wiki/Q3798668","display_name":"Aerospace engineering","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.5951/mt.65.5.0431","doi":"https://doi.org/10.5951/mt.65.5.0431","is_oa":false,"landing_page_url":"https://doi.org/10.5951/mt.65.5.0431","pdf_url":null,"source":{"id":"https://openalex.org/S2764739392","display_name":"Mathematics Teacher Learning and Teaching PK-12","issn_l":"0025-5769","issn":["0025-5769","2330-0582"],"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/P4310315778","host_organization_name":"National Council of Teachers of Mathematics","host_organization_lineage":["https://openalex.org/P4310315778"],"host_organization_lineage_names":["National Council of Teachers of Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Mathematics Teacher","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W275929964","https://openalex.org/W2107773297","https://openalex.org/W4388445250","https://openalex.org/W2043013420","https://openalex.org/W7219545215","https://openalex.org/W427235362","https://openalex.org/W4238444299","https://openalex.org/W3119242157","https://openalex.org/W4391562360","https://openalex.org/W2799547563"],"abstract_inverted_index":{"A":[0,45],"visual":[1],"aid":[2,32],"for":[3,56],"vividly":[4],"demonstrating":[5],"the":[6,9,31,36,76,89,92,96],"relationship":[7],"of":[8,23,47,91],"direct":[10],"and":[11,20,30,71,78,82,99],"inverse":[12,93],"circular":[13],"functions":[14],"can":[15],"be":[16],"made":[17],"quite":[18],"easily":[19],"cheaply":[21],"out":[22],"transparent":[24,54],"sheets.":[25],"The":[26,85],"principle":[27],"is":[28,33,51,67],"simple,":[29],"similar":[34],"to":[35],"graphs":[37],"that":[38,59,75],"are":[39,80],"shown":[40],"in":[41],"all":[42],"trigonometry":[43],"texts.":[44],"graph":[46,66,90],"a":[48,62],"trigonometric":[49],"function":[50,94],"drawn":[52],"on":[53],"material":[55],"an":[57],"interval":[58],"makes":[60],"it":[61],"one-to-one":[63],"function.":[64],"This":[65],"then":[68],"turned":[69],"over":[70],"rotated":[72],"90\u00b0,":[73],"so":[74],"x-":[77],"y-axes":[79],"interchanged":[81],"properly":[83],"directed.":[84],"viewer":[86],"now":[87],"sees":[88],"with":[95],"proper":[97],"domain":[98],"range.":[100]},"counts_by_year":[],"updated_date":"2026-08-21T09:56:20.448147","created_date":"2016-06-24T00:00:00"}