{"id":1914,"date":"2020-06-12T07:28:57","date_gmt":"2020-06-12T07:28:57","guid":{"rendered":"http:\/\/ibalmaths.com\/?post_type=knowledgebase&#038;p=1914"},"modified":"2020-06-12T08:39:01","modified_gmt":"2020-06-12T08:39:01","slug":"practice-questions-trigonometric-identities-and-equations","status":"publish","type":"knowledgebase","link":"https:\/\/ibalmaths.com\/index.php\/ibdp-math-hl-2\/trigonometric-identities-and-equations\/practice-questions-trigonometric-identities-and-equations\/","title":{"rendered":"Practice Questions &#8211; Trigonometric Identities and Equations"},"content":{"rendered":"<p>1. (i) Prove the identity\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfenced><mrow><mi>sin<\/mi><mi>&#952;<\/mi><mo>+<\/mo><mi>cos<\/mi><mi>&#952;<\/mi><\/mrow><\/mfenced><mfenced><mrow><mn>1<\/mn><mo>&#8211;<\/mo><mi>sin<\/mi><mi>&#952;<\/mi><mi>cos<\/mi><mi>&#952;<\/mi><\/mrow><\/mfenced><mo>=<\/mo><msup><mi>sin<\/mi><mn>3<\/mn><\/msup><mi>&#952;<\/mi><mo>&#8211;<\/mo><msup><mi>cos<\/mi><mn>3<\/mn><\/msup><mi>&#952;<\/mi><\/math>\r\n .\u00a0 \u00a0[3 marks]<\/p>\r\n<p>(ii) Hence solve the equation \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfenced><mrow><mi>sin<\/mi><mi>&#952;<\/mi><mo>+<\/mo><mi>cos<\/mi><mi>&#952;<\/mi><\/mrow><\/mfenced><mfenced><mrow><mn>1<\/mn><mo>&#8211;<\/mo><mi>sin<\/mi><mi>&#952;<\/mi><mi>cos<\/mi><mi>&#952;<\/mi><\/mrow><\/mfenced><mo>=<\/mo><mn>3<\/mn><msup><mi>cos<\/mi><mn>3<\/mn><\/msup><mi>&#952;<\/mi><\/math>\r\n , for\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>0<\/mn><mo>&#176;<\/mo><mo>&#8804;<\/mo><mi>&#952;<\/mi><mo>&#8804;<\/mo><mn>360<\/mn><mo>&#176;<\/mo><\/math>\r\n .\u00a0 \u00a0[3 marks]<\/p>\r\n<div id=\"link1-link-1914\" class=\"sh-link link1-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link1', 1914, 'Solution1', 'Hide Solution1'); return false;\" aria-expanded=\"false\"><span id=\"link1-toggle-1914\">Solution1<\/span><\/a><\/div><div id=\"link1-content-1914\" class=\"sh-content link1-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1915\" src=\"http:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/06\/1.png\" alt=\"\" width=\"622\" height=\"168\" srcset=\"https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/06\/1.png 622w, https:\/\/ibalmaths.com\/wp-content\/uploads\/2020\/06\/1-300x81.png 300w\" sizes=\"auto, (max-width: 622px) 100vw, 622px\" \/><\/p>\r\n<p>\u00a0<\/div>\r\n<p>2. (i) Express\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>5<\/mn><mo>&#160;<\/mo><mi>cos<\/mi><mi>&#952;<\/mi><mo>&#8211;<\/mo><mn>2<\/mn><mo>&#160;<\/mo><mi>sin<\/mi><mi>&#952;<\/mi><\/math>\r\n in the form \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>R<\/mi><mi>cos<\/mi><mfenced><mrow><mi>&#952;<\/mi><mo>+<\/mo><mi>&#945;<\/mi><\/mrow><\/mfenced><\/math>\r\n , where \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>R<\/mi><mo>&#62;<\/mo><mn>0<\/mn><\/math>\r\n and\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>0<\/mn><mo>&#60;<\/mo><mi>&#945;<\/mi><mo>&#60;<\/mo><mfrac><mi>&#960;<\/mi><mn>2<\/mn><\/mfrac><\/math>\r\n . Give the value of\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>&#945;<\/mi><\/math>\r\n correct to 4 decimal places. [3 marks]<\/p>\r\n<p>(ii) Using your answer from part (i), solve the equation\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>5<\/mn><mo>&#160;<\/mo><mi>c<\/mi><mi>o<\/mi><mi>t<\/mi><mi>&#952;<\/mi><mo>&#8211;<\/mo><mn>4<\/mn><mo>&#160;<\/mo><mi>cos<\/mi><mi>e<\/mi><mi>c<\/mi><mi>&#952;<\/mi><mo>=<\/mo><mn>2<\/mn><\/math>\r\n for\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>0<\/mn><mo>&#60;<\/mo><mi>&#952;<\/mi><mo>&#60;<\/mo><mn>2<\/mn><mi>&#960;<\/mi><\/math>\r\n .\u00a0 \u00a0[5 marks]<\/p>\r\n<div id=\"link2-link-1914\" class=\"sh-link link2-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link2', 1914, 'Solution2', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link2-toggle-1914\">Solution2<\/span><\/a><\/div><div id=\"link2-content-1914\" class=\"sh-content link2-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p>(i)\u00a0\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>R<\/mi><mo>=<\/mo><msqrt><mn>29<\/mn><\/msqrt><mo>&#160;<\/mo><mi>o<\/mi><mi>r<\/mi><mo>&#160;<\/mo><mn>5<\/mn><mo>.<\/mo><mn>39<\/mn><\/math>\r\n \u00a0 \u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>&#945;<\/mi><mo>=<\/mo><mn>0<\/mn><mo>.<\/mo><mn>3805<\/mn><\/math>\r\n<\/p>\r\n<p>(ii) 0.353, 5.17<\/p>\r\n<p>\u00a0<\/div>\r\n<p>3. (i) Prove the identity\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfrac><mrow><mn>1<\/mn><mo>+<\/mo><mi>cos<\/mi><mi>&#952;<\/mi><\/mrow><mrow><mi>sin<\/mi><mi>&#952;<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mfrac><mrow><mi>sin<\/mi><mi>&#952;<\/mi><\/mrow><mrow><mn>1<\/mn><mo>+<\/mo><mi>cos<\/mi><mi>&#952;<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>2<\/mn><mrow><mi>sin<\/mi><mi>&#952;<\/mi><\/mrow><\/mfrac><\/math>\r\n . [3 marks]<\/p>\r\n<p>(ii) Hence solve the equation\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfrac><mrow><mn>1<\/mn><mo>+<\/mo><mi>cos<\/mi><mi>&#952;<\/mi><\/mrow><mrow><mi>sin<\/mi><mi>&#952;<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mfrac><mrow><mi>sin<\/mi><mi>&#952;<\/mi><\/mrow><mrow><mn>1<\/mn><mo>+<\/mo><mi>cos<\/mi><mi>&#952;<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>3<\/mn><mrow><mi>cos<\/mi><mi>&#952;<\/mi><\/mrow><\/mfrac><\/math>\r\n for\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>0<\/mn><mo>&#176;<\/mo><mo>&#8804;<\/mo><mi>&#952;<\/mi><mo>&#8804;<\/mo><mn>360<\/mn><mo>&#176;<\/mo><\/math>\r\n . [4 marks]<\/p>\r\n<div id=\"link3-link-1914\" class=\"sh-link link3-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link3', 1914, 'Solution3', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link3-toggle-1914\">Solution3<\/span><\/a><\/div><div id=\"link3-content-1914\" class=\"sh-content link3-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p>(ii) \u03b8 = 33.7\u00b0 or 213.7\u00b0<\/p>\r\n<p>\u00a0<\/div>\r\n<p>4. (i) Express\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>2<\/mn><mo>&#160;<\/mo><mi>cos<\/mi><mi>&#952;<\/mi><mo>+<\/mo><msqrt><mn>5<\/mn><\/msqrt><mo>&#160;<\/mo><mi>sin<\/mi><mi>&#952;<\/mi><\/math>\r\n in the form \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>R<\/mi><mi>cos<\/mi><mfenced><mrow><mi>&#952;<\/mi><mo>&#8211;<\/mo><mi>&#945;<\/mi><\/mrow><\/mfenced><\/math>\r\n where R &gt; 0 and \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>0<\/mn><mo>&#176;<\/mo><mo>&#60;<\/mo><mi>&#945;<\/mi><mo>&#60;<\/mo><mn>90<\/mn><mo>&#176;<\/mo><\/math>\r\n , giving the value of\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>&#945;<\/mi><\/math>\r\n correct to 2 decimal places. [3 marks]<\/p>\r\n<p>(ii) Hence solve the equation\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>2<\/mn><mo>&#160;<\/mo><mi>cos<\/mi><mi>&#952;<\/mi><mo>+<\/mo><msqrt><mn>5<\/mn><\/msqrt><mo>&#160;<\/mo><mi>sin<\/mi><mi>&#952;<\/mi><mo>=<\/mo><mn>1<\/mn><\/math>\r\n for\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>0<\/mn><mo>&#176;<\/mo><mo>&#60;<\/mo><mi>&#952;<\/mi><mo>&#60;<\/mo><mn>360<\/mn><mo>&#176;<\/mo><\/math>\r\n .\u00a0 \u00a0[4 marks]<\/p>\r\n<div id=\"link4-link-1914\" class=\"sh-link link4-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('link4', 1914, 'Solution4', 'Hide Solution'); return false;\" aria-expanded=\"false\"><span id=\"link4-toggle-1914\">Solution4<\/span><\/a><\/div><div id=\"link4-content-1914\" class=\"sh-content link4-content sh-hide\" style=\"display: none;\"><\/p>\r\n<p>(i)\u00a0 \r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>R<\/mi><mo>=<\/mo><mn>3<\/mn><mo>,<\/mo><mo>&#160;<\/mo><mo>&#160;<\/mo><mi>&#945;<\/mi><mo>=<\/mo><mn>48<\/mn><mo>.<\/mo><mn>19<\/mn><\/math>\r\n<\/p>\r\n<p>(ii) 118.7, 337.7<\/p>\r\n<p>\u00a0<\/div><!-- AddThis Advanced Settings generic via filter on the_content --><!-- AddThis Share Buttons generic via filter on the_content -->","protected":false},"excerpt":{"rendered":"1. (i) Prove the identity\u00a0 sin&#952;+cos&#952;1&#8211;sin&#952;cos&#952;=sin3&#952;&#8211;cos3&#952; .\u00a0 \u00a0[3 marks] (ii) Hence solve the equation sin&#952;+cos&#952;1&#8211;sin&#952;cos&#952;=3cos3&#952; , for\u00a0 0&#176;&#8804;&#952;&#8804;360&#176; .\u00a0 \u00a0[3 marks] 2. (i) Express\u00a0 5&#160;cos&#952;&#8211;2&#160;sin&#952; in the form Rcos&#952;+&#945; , where R&#62;0 and\u00a0 0&#60;&#945;&#60;&#960;2 . Give the value of\u00a0 &#945; correct to 4 decimal places. [3 marks] (ii) Using your answer from part (i), solve [&hellip;]<!-- AddThis Advanced Settings generic via filter on get_the_excerpt --><!-- AddThis Share Buttons generic via filter on get_the_excerpt -->","protected":false},"author":4,"featured_media":0,"comment_status":"closed","ping_status":"closed","template":"","class_list":["post-1914","knowledgebase","type-knowledgebase","status-publish","hentry","knowledgebase_cat-trigonometric-identities-and-equations","no-wpautop"],"_links":{"self":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1914","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase"}],"about":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/types\/knowledgebase"}],"author":[{"embeddable":true,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/comments?post=1914"}],"version-history":[{"count":8,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1914\/revisions"}],"predecessor-version":[{"id":1932,"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/knowledgebase\/1914\/revisions\/1932"}],"wp:attachment":[{"href":"https:\/\/ibalmaths.com\/index.php\/wp-json\/wp\/v2\/media?parent=1914"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}