{"id":2755,"date":"2025-08-13T18:35:39","date_gmt":"2025-08-13T18:35:39","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2755"},"modified":"2025-10-17T21:41:39","modified_gmt":"2025-10-17T21:41:39","slug":"triangle-trigonometry-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/triangle-trigonometry-background-youll-need-1\/","title":{"raw":"Triangle Trigonometry: Background You'll Need 1","rendered":"Triangle Trigonometry: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Use the pythagorean theorem to solve right triangles<\/span><\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>Pythagorean Theorem<\/h3>\r\n<p data-start=\"422\" data-end=\"526\">In a right triangle, the sides have a special relationship described by the Pythagorean Theorem:<\/p>\r\n<p data-start=\"528\" data-end=\"560\">[latex]a^2 + b^2 = c^2[\/latex]<\/p>\r\n<p data-start=\"562\" data-end=\"695\">where [latex]a[\/latex] and [latex]b[\/latex] are the legs, and [latex]c[\/latex] is the hypotenuse (the side opposite the right angl<\/p>\r\n\r\n<\/section>\r\n<p data-start=\"697\" data-end=\"792\">This relationship lets us find any missing side of a right triangle when we know the other two.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p data-start=\"814\" data-end=\"910\">Find the hypotenuse of a right triangle with legs [latex]a = 6[\/latex] and [latex]b = 8[\/latex].<\/p>\r\n<p data-start=\"912\" data-end=\"1067\">[latex]\r\n\\begin{align}\r\na^2 + b^2 &amp;= c^2 \\[4pt]\r\n6^2 + 8^2 &amp;= c^2 \\[4pt]\r\n36 + 64 &amp;= c^2 \\[4pt]\r\n100 &amp;= c^2 \\[4pt]\r\nc &amp;= \\sqrt{100} = 10\r\n\\end{align}\r\n[\/latex]<\/p>\r\n<p data-start=\"1069\" data-end=\"1098\">Result: [latex]c = 10[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p data-start=\"1120\" data-end=\"1191\">Find a missing leg when [latex]b = 9[\/latex] and [latex]c = 15[\/latex].<\/p>\r\n<p data-start=\"1193\" data-end=\"1337\">[latex]\r\n\\begin{align}\r\na^2 + b^2 &amp;= c^2 \\[4pt]\r\na^2 + 9^2 &amp;= 15^2 \\[4pt]\r\na^2 + 81 &amp;= 225 \\[4pt]\r\na^2 &amp;= 144 \\[4pt]\r\na &amp;= 12\r\n\\end{align}\r\n[\/latex]<\/p>\r\n<p data-start=\"1339\" data-end=\"1368\">Result: [latex]a = 12[\/latex][ohm_question hide_question_numbers=1]313784[\/ohm_question]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]313785[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]313784[\/ohm_question]<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">The hypotenuse is always the longest side and is opposite the right angle.<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Use the pythagorean theorem to solve right triangles<\/span><\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>Pythagorean Theorem<\/h3>\n<p data-start=\"422\" data-end=\"526\">In a right triangle, the sides have a special relationship described by the Pythagorean Theorem:<\/p>\n<p data-start=\"528\" data-end=\"560\">[latex]a^2 + b^2 = c^2[\/latex]<\/p>\n<p data-start=\"562\" data-end=\"695\">where [latex]a[\/latex] and [latex]b[\/latex] are the legs, and [latex]c[\/latex] is the hypotenuse (the side opposite the right angl<\/p>\n<\/section>\n<p data-start=\"697\" data-end=\"792\">This relationship lets us find any missing side of a right triangle when we know the other two.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p data-start=\"814\" data-end=\"910\">Find the hypotenuse of a right triangle with legs [latex]a = 6[\/latex] and [latex]b = 8[\/latex].<\/p>\n<p data-start=\"912\" data-end=\"1067\">[latex]\\begin{align}  a^2 + b^2 &= c^2 \\[4pt]  6^2 + 8^2 &= c^2 \\[4pt]  36 + 64 &= c^2 \\[4pt]  100 &= c^2 \\[4pt]  c &= \\sqrt{100} = 10  \\end{align}[\/latex]<\/p>\n<p data-start=\"1069\" data-end=\"1098\">Result: [latex]c = 10[\/latex]<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p data-start=\"1120\" data-end=\"1191\">Find a missing leg when [latex]b = 9[\/latex] and [latex]c = 15[\/latex].<\/p>\n<p data-start=\"1193\" data-end=\"1337\">[latex]\\begin{align}  a^2 + b^2 &= c^2 \\[4pt]  a^2 + 9^2 &= 15^2 \\[4pt]  a^2 + 81 &= 225 \\[4pt]  a^2 &= 144 \\[4pt]  a &= 12  \\end{align}[\/latex]<\/p>\n<p data-start=\"1339\" data-end=\"1368\">Result: [latex]a = 12[\/latex]<iframe loading=\"lazy\" id=\"ohm313784\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313784&theme=lumen&iframe_resize_id=ohm313784&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm313785\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313785&theme=lumen&iframe_resize_id=ohm313785&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm313784\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313784&theme=lumen&iframe_resize_id=ohm313784&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">The hypotenuse is always the longest side and is opposite the right angle.<\/section>\n","protected":false},"author":67,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":221,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2755"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2755\/revisions"}],"predecessor-version":[{"id":4750,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2755\/revisions\/4750"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/221"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2755\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2755"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2755"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2755"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2755"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}