{"id":248,"date":"2017-08-23T10:32:15","date_gmt":"2017-08-23T10:32:15","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=248"},"modified":"2017-08-31T05:41:45","modified_gmt":"2017-08-31T05:41:45","slug":"half-angle-and-double-angle-formulas","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/half-angle-and-double-angle-formulas\/","title":{"rendered":"Half-Angle and Double-Angle Formulas"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/graphing-trigonometric-functions-and-inverse-trigonometric-functions\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/trigonometry\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/rectangular-and-polar-coordinates\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Half-Angle and Double-Angle Formulas<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, we will define and learn to apply addition, half-angle, and double-angle formulas.<\/p>\n<h4>Previously Covered:<\/h4>\n<p>There are six <strong><em>trigonometric ratios <\/em><\/strong>that\u00a0can help you to solve for lengths of sides in right triangles.<\/p>\n<ul>\n<li><em><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p1_clip_image002.gif\" width=\"127\" height=\"44\" name=\"graphics2\" align=\"ABSMIDDLE\" border=\"0\" \/><\/em><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p1_clip_image004.gif\" width=\"129\" height=\"44\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p1_clip_image006.gif\" width=\"112\" height=\"44\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p1_clip_image008.gif\" width=\"112\" height=\"44\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p1_clip_image010.gif\" width=\"128\" height=\"44\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/><\/strong><\/li>\n<li><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p1_clip_image012.gif\" width=\"128\" height=\"44\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/><\/strong><\/li>\n<\/ul>\n<section>\n<h3>What are the addition formulas?<\/h3>\n<p>In an earlier section, we mentioned one of the addition\u00a0formulas. The <strong>addition formulas<\/strong> are another group\u00a0of identities that are useful tools for trigonometry. If you\u00a0memorize these formulas, you can use information from the standard\u00a0points you learned earlier to find uncommon points without using a\u00a0calculator.<\/p>\n<p>The sine and cosine addition formulas are:<\/p>\n<ul>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p2_clip_image002.gif\" width=\"257\" height=\"23\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p2_clip_image004.gif\" width=\"261\" height=\"21\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p2_clip_image006.gif\" width=\"257\" height=\"21\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p2_clip_image008.gif\" width=\"261\" height=\"21\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<\/ul>\n<p>Using these sine and cosine formulas, you can easily derive the\u00a0tangent, cotangent, secant, and cosecant addition formulas.\u00a0Together, we will derive the formula for \u00a0<em><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p2_clip_image010.gif\" width=\"68\" height=\"21\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0<\/strong><\/em>in order to get started in our effort to derive the other addition\u00a0formulas. Pay close attention, because you will be asked to derive\u00a0one of the addition formulas not given in this section at the end\u00a0of this module.<\/p>\n<p>Using the addition formulas above, what is the formula for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p2_clip_image012.gif\" width=\"71\" height=\"21\" name=\"graphics8\" align=\"TEXTTOP\" border=\"0\" \/>?<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p2_clip_image014.gif\" width=\"151\" height=\"44\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p2_clip_image016.gif\" width=\"196\" height=\"44\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p2_clip_image018.gif\" width=\"381\" height=\"44\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p2_clip_image020.gif\" width=\"199\" height=\"88\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p2_clip_image022.gif\" width=\"199\" height=\"88\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p2_clip_image024.gif\" width=\"256\" height=\"92\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p2_clip_image026.gif\" width=\"121\" height=\"44\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>So the addition formula for tangent is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p2_clip_image028.gif\" width=\"188\" height=\"44\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>Use the sine and cosine addition formulas to prove that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p2_clip_image030.gif\" width=\"188\" height=\"44\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0The method is the same as the one above.<\/p>\n<p><em>Challenge<\/em> : Derive the cotangent, secant, and cosecant\u00a0addition formulas that have not been given above.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is cos(75\u00b0)?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p2_clip_image032.gif\" width=\"35\" height=\"44\" name=\"graphics18\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p2_clip_image034.gif\" width=\"47\" height=\"48\" name=\"graphics19\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p2_clip_image036.gif\" width=\"27\" height=\"45\" name=\"graphics20\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li>2<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is B.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p2_clip_image038.gif\" width=\"173\" height=\"24\" name=\"graphics21\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>And using the cosine addition formula you get<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p2_clip_image040.gif\" width=\"243\" height=\"24\" name=\"graphics22\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Using the standard points above, you get<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p2_clip_image042.gif\" width=\"124\" height=\"96\" name=\"graphics23\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<h3>What are the double-angle formulas?<\/h3>\n<p>In addition to addition formulas, double-angle formulas are\u00a0also useful when you are trying to use information from the\u00a0standard point to find uncommon points without using a calculator.\u00a0The basic <strong>double-angle formulas<\/strong> are:<\/p>\n<ul>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image002.gif\" width=\"159\" height=\"21\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image004.gif\" width=\"164\" height=\"24\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image006.gif\" width=\"88\" height=\"21\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image008.gif\" width=\"85\" height=\"21\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image010.gif\" width=\"132\" height=\"41\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<\/ul>\n<p>There are other double-angle formulas, but these three are the\u00a0most important, and the only double angle formulas that we will\u00a0cover at this time.<\/p>\n<p><strong><span style=\"text-decoration: none;\">Can you prove that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image011.gif\" width=\"159\" height=\"21\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0using the addition formulas? <\/span><\/strong><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image013.gif\" width=\"257\" height=\"23\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><span style=\"text-decoration: none;\">Let <\/span><em><span style=\"text-decoration: none;\">b\u00a0<\/span><\/em><span style=\"text-decoration: none;\">= <\/span><em><span style=\"text-decoration: none;\">a<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0and use substitution to get<\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image015.gif\" width=\"260\" height=\"21\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p>Now pull out the sin<span style=\"text-decoration: none;\">(<\/span><em><span style=\"text-decoration: none;\">a<\/span><\/em><span style=\"text-decoration: none;\">)<\/span>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image017.gif\" width=\"213\" height=\"21\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image019.gif\" width=\"167\" height=\"23\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image020.gif\" width=\"159\" height=\"21\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><strong><span style=\"text-decoration: none;\">Can you prove the\u00a0double-angle formula\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image021.gif\" width=\"164\" height=\"24\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image022.gif\" width=\"88\" height=\"21\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image023.gif\" width=\"85\" height=\"21\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/>?<\/span><\/strong><\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image025.gif\" width=\"261\" height=\"21\" name=\"graphics17\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><span style=\"text-decoration: none;\">Let <\/span><em><span style=\"text-decoration: none;\">b\u00a0<\/span><\/em><span style=\"text-decoration: none;\">= <\/span><em><span style=\"text-decoration: none;\">a,\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and<\/span> use substitution to get<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image027.gif\" width=\"263\" height=\"21\" name=\"graphics18\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image028.gif\" width=\"164\" height=\"24\" name=\"graphics19\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Since<\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image030.gif\" width=\"120\" height=\"21\" name=\"graphics20\" align=\"BOTTOM\" border=\"0\" \/>,<\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image032.gif\" width=\"120\" height=\"21\" name=\"graphics21\" align=\"BOTTOM\" border=\"0\" \/><br \/>\n.<\/p>\n<p>Use substitution to get<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image034.gif\" width=\"193\" height=\"24\" name=\"graphics22\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image035.gif\" width=\"88\" height=\"21\" name=\"graphics23\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Using\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image036.gif\" width=\"120\" height=\"21\" name=\"graphics24\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image038.gif\" width=\"120\" height=\"21\" name=\"graphics25\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0use substitution to get<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image040.gif\" width=\"99\" height=\"24\" name=\"graphics26\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image042.gif\" width=\"108\" height=\"21\" name=\"graphics27\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image043.gif\" width=\"85\" height=\"21\" name=\"graphics28\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><strong><span style=\"text-decoration: none;\">Can you prove the\u00a0double-angle formula\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image044.gif\" width=\"132\" height=\"41\" name=\"graphics29\" align=\"ABSMIDDLE\" border=\"0\" \/>?<br \/>\n<\/span><\/strong><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image046.gif\" width=\"188\" height=\"44\" name=\"graphics30\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>No<span style=\"text-decoration: none;\">w let <\/span><em><span style=\"text-decoration: none;\">b\u00a0= a<\/span><\/em><span style=\"text-decoration: none;\"> to get <\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image048.gif\" width=\"189\" height=\"44\" name=\"graphics31\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image049.gif\" width=\"132\" height=\"41\" name=\"graphics32\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is tan(120\u00b0)?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image051.gif\" width=\"35\" height=\"23\" name=\"graphics33\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li>2<\/li>\n<li>\u20133<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image053.gif\" width=\"36\" height=\"24\" name=\"graphics34\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is D.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image055.gif\" width=\"157\" height=\"51\" name=\"graphics35\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Using the tangent double-angle formula,<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image057.gif\" width=\"95\" height=\"44\" name=\"graphics36\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p>And from the standard points we learned above, we know that<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p3_clip_image059.gif\" width=\"79\" height=\"133\" name=\"graphics37\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<h3>What are half-angle formulas?<\/h3>\n<p><strong>Half-angle formulas<\/strong> are more common than any\u00a0other type of identity used in trigonometry. Yes, more identities!\u00a0All of the identities we have learned, including the half-angle\u00a0formulas, are still not the entirety of the trigonometric\u00a0identities. The more identities you know, the easier it will be\u00a0for you to recognize the simplified solution to a complex\u00a0expression. All of these identities can be used to simplify more\u00a0complex equations, and can be used to solve for angles using\u00a0angles that you already know.<\/p>\n<p>The half-angle formulas are:<\/p>\n<ul>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p4_clip_image002.gif\" width=\"143\" height=\"48\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p4_clip_image004.gif\" width=\"145\" height=\"48\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p4_clip_image006.gif\" width=\"145\" height=\"48\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<\/ul>\n<p>As you have seen, you can use other identities to derive new\u00a0ones. The same is true for the half-angle formulas. The rest of\u00a0this section will cover methods of deriving these formulas, but\u00a0try and do them yourself before moving ahead. Try using one of the\u00a0identities that you learned in this module to derive these\u00a0half-angle formulas.<\/p>\n<p>Did you try to derive the half-angle formulas yourself?<\/p>\n<p>In order to prove that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p4_clip_image007.gif\" width=\"143\" height=\"48\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0use the double-angle formula<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p4_clip_image009.gif\" width=\"164\" height=\"24\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p4_clip_image011.gif\" width=\"85\" height=\"21\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p>You can rearrange this formula to get<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p4_clip_image013.gif\" width=\"139\" height=\"24\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Now let\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p4_clip_image015.gif\" width=\"41\" height=\"41\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and use substitution to get<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/s8_p4_html_m3a538d6e.gif\" width=\"169\" height=\"45\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Divide both sides by 2.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p4_clip_image019.gif\" width=\"139\" height=\"45\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Now, take the square root of both sides.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p4_clip_image021.gif\" width=\"163\" height=\"51\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p4_clip_image023.gif\" width=\"152\" height=\"48\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>In order to prove that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p4_clip_image024.gif\" width=\"145\" height=\"48\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0use the double-angle formula.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p4_clip_image025.gif\" width=\"164\" height=\"24\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p4_clip_image027.gif\" width=\"88\" height=\"21\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p>If you did not prove these identities before, stop now and try\u00a0to prove this one before moving ahead. This method is exactly the\u00a0same as the one above.<\/p>\n<p>Let\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p4_clip_image029.gif\" width=\"41\" height=\"41\" name=\"graphics18\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Use substitution.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p4_clip_image031.gif\" width=\"175\" height=\"45\" name=\"graphics19\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Add 1 to both sides.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p4_clip_image033.gif\" width=\"147\" height=\"45\" name=\"graphics20\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Divide both sides by 2.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p4_clip_image035.gif\" width=\"141\" height=\"45\" name=\"graphics21\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Now take the square root of both sides.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/s8_p4_html_m6f3bb2c8.gif\" width=\"157\" height=\"99\" name=\"graphics22\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Now, prove the half-angle formula\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p5_clip_image002.gif\" width=\"145\" height=\"48\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0using the sine and cosine half-angle formulas.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p5_clip_image004.gif\" width=\"164\" height=\"232\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is sin (15\u00b0) ?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p5_clip_image006.gif\" width=\"81\" height=\"49\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li>3<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p5_clip_image008.gif\" width=\"27\" height=\"45\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li>2<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is A.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p5_clip_image010.gif\" width=\"135\" height=\"51\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Use the sine half-angle formula to get<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p5_clip_image012.gif\" width=\"119\" height=\"48\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p>Since you know that cos(30\u00b0)=\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p5_clip_image013.gif\" width=\"27\" height=\"45\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0because it is one of the standard points that you memorized, we\u00a0get<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p5_clip_image015.gif\" width=\"103\" height=\"117\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<\/div>\n<\/section>\n<h3>Review of New Vocabulary and Concepts<\/h3>\n<p>The <strong><em>addition formulas<\/em><\/strong> are:<\/p>\n<ul>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p6_clip_image002.gif\" width=\"257\" height=\"23\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p6_clip_image004.gif\" width=\"261\" height=\"21\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p6_clip_image006.gif\" width=\"257\" height=\"21\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p6_clip_image008.gif\" width=\"261\" height=\"21\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p6_clip_image010.gif\" width=\"188\" height=\"44\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p6_clip_image012.gif\" width=\"188\" height=\"44\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<\/ul>\n<p>The basic <em><strong>double-angle formulas<\/strong><\/em> are:<\/p>\n<ul>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p6_clip_image014.gif\" width=\"159\" height=\"21\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p6_clip_image016.gif\" width=\"164\" height=\"24\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p6_clip_image018.gif\" width=\"88\" height=\"21\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p6_clip_image020.gif\" width=\"85\" height=\"21\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p6_clip_image022.gif\" width=\"132\" height=\"41\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<\/ul>\n<p>The <em><strong>half-angle formulas<\/strong><\/em> are:<\/p>\n<ul>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p6_clip_image024.gif\" width=\"143\" height=\"48\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p6_clip_image026.gif\" width=\"145\" height=\"48\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s5_p6_clip_image028.gif\" width=\"145\" height=\"48\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<\/ul>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/graphing-trigonometric-functions-and-inverse-trigonometric-functions\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/trigonometry\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/rectangular-and-polar-coordinates\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Half-Angle and Double-Angle Formulas Objective In this lesson, we will define and learn to apply addition, half-angle, and double-angle formulas. Previously Covered: There are six trigonometric ratios that\u00a0can help you to solve for lengths of sides in right triangles. What are the addition formulas? In an earlier section, we [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-248","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/248","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=248"}],"version-history":[{"count":6,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/248\/revisions"}],"predecessor-version":[{"id":616,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/248\/revisions\/616"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=248"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}