{"id":2197,"date":"2023-05-13T16:23:10","date_gmt":"2023-05-13T20:23:10","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=2197"},"modified":"2024-10-25T12:09:05","modified_gmt":"2024-10-25T16:09:05","slug":"trigonometry-challenge-1-%e2%ad%90%e2%ad%90","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=2197","title":{"rendered":"Trigonometry Challenge &#8211; 1 \u2b50\u2b50"},"content":{"rendered":"\n<p>Prove that $\\ \\ \\ \\ \\ \\ \\ \\dfrac{sin 20^\\circ}{cos 20^\\circ-2\\cdot sin 10^\\circ}=tan 30^\\circ\\ \\ \\ \\ \\ \\ \\ \\ \\ $ <a href=\"javascript:toggle_visibility('tri-chall-1')\">Solution<\/a><div id='tri-chall-1' style='display:none'><\/p>\n\n\n\n<p><strong>Proof<\/strong>: Because $$ sin 10^\\circ-sin 10^\\circ=0,\\ \\ \\ sin^2 20^\\circ+cos^2 20^\\circ=1,\\ \\ \\ sin 30^\\circ =\\frac{1}{2}$$<\/p>\n\n\n\n<p>We have $$(sin^2 20^\\circ+cos^2 20^\\circ)\\cdot sin 10^\\circ-2\\cdot sin 10^\\circ\\cdot sin 30^\\circ=0$$<\/p>\n\n\n\n<p>Adding $\\ sin 20^\\circ\\cdot cos 20^\\circ\\cdot cos 10^\\circ\\ $ to both sides of the above equation, we have: <\/p>\n\n\n\n<p>$$ sin 20^\\circ\\cdot cos 20^\\circ\\cdot cos 10^\\circ + (sin^2 20^\\circ+cos^2 20^\\circ)\\cdot sin 10^\\circ-2\\cdot sin 10^\\circ\\cdot sin 30^\\circ$$<\/p>\n\n\n\n<p>$$=sin 20^\\circ\\cdot cos 20^\\circ\\cdot cos 10^\\circ $$<\/p>\n\n\n\n<p>Moving the term $\\ sin^2 20^\\circ\\cdot sin 10^\\circ\\ $ from the left side to the right side of the above equation, we have:<\/p>\n\n\n\n<p>$$ sin 20^\\circ\\cdot cos 20^\\circ\\cdot cos 10^\\circ + cos^2 20^\\circ\\cdot sin 10^\\circ-2\\cdot sin 10^\\circ\\cdot sin 30^\\circ$$<\/p>\n\n\n\n<p>$$=sin 20^\\circ\\cdot cos 20^\\circ\\cdot cos 10^\\circ-sin^2 20^\\circ\\cdot sin 10^\\circ$$<\/p>\n\n\n\n<p>Combining the first two terms on the left side and two terms on the right side of the above equation, we have:<\/p>\n\n\n\n<p>$$ cos 20^\\circ\\cdot(sin 20^\\circ\\cdot cos 10^\\circ+cos 20^\\circ\\cdot sin 10^\\circ)-2\\cdot sin 10^\\circ\\cdot sin 30^\\circ$$<\/p>\n\n\n\n<p>$$=sin 20^\\circ\\cdot(cos 20^\\circ\\cdot cos 10^\\circ-sin 20^\\circ\\cdot sin 10^\\circ)$$<\/p>\n\n\n\n<p>Because $$ sin 30^\\circ=sin(20^\\circ+10^\\circ)=sin 20^\\circ\\cdot cos 10^\\circ+cos 20^\\circ\\cdot sin 10^\\circ $$<\/p>\n\n\n\n<p>$$ cos 30^\\circ=cos(20^\\circ+10^\\circ)=cos 20^\\circ\\cdot cos 10^\\circ-sin 20^\\circ\\cdot sin 10^\\circ$$<\/p>\n\n\n\n<p>We have: $$ cos 20^\\circ\\cdot sin 30^\\circ-2\\cdot sin 10^\\circ\\cdot sin 30^\\circ=sin 20^\\circ\\cdot cos 30^\\circ$$<\/p>\n\n\n\n<p>Finally, $$\\boxed{\\dfrac{sin 20^\\circ}{cos 20^\\circ-2\\cdot sin 10^\\circ}=\\dfrac{sin 30^\\circ}{cos 30^\\circ}=tan 30^\\circ}$$<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Prove that $\\ \\ \\ \\ \\ \\ \\ \\dfrac{sin 20^\\circ}{cos 20^\\circ-2\\cdot sin 10^\\circ}=tan 30^\\circ\\ \\ \\ \\ \\ \\ \\ \\ \\ $ Solution Proof: Because $$ sin 10^\\circ-sin 10^\\circ=0,\\ \\ \\ sin^2 20^\\circ+cos^2 20^\\circ=1,\\ \\ \\ sin 30^\\circ &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=2197\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[14,15],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2197"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2197"}],"version-history":[{"count":38,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2197\/revisions"}],"predecessor-version":[{"id":2236,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2197\/revisions\/2236"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2197"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2197"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2197"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}