{"id":46473,"date":"2025-07-02T06:45:28","date_gmt":"2025-07-02T06:45:28","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=46473"},"modified":"2025-07-02T06:45:29","modified_gmt":"2025-07-02T06:45:29","slug":"lx-sin7pi-2x-7pi-2-cos7pi-2","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/lx-sin7pi-2x-7pi-2-cos7pi-2\/","title":{"rendered":"L(x) = -sin(7pi\/2)(x &#8211; 7pi\/2) + cos(7pi\/2)"},"content":{"rendered":"\n<p>L(x) = -sin(7pi\/2)(x &#8211; 7pi\/2) + cos(7pi\/2)<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>The function you&#8217;ve provided is:L(x)=\u2212sin\u2061(7\u03c02)(x\u22127\u03c02)+cos\u2061(7\u03c02)L(x) = -\\sin\\left(\\frac{7\\pi}{2}\\right) \\left( x &#8211; \\frac{7\\pi}{2} \\right) + \\cos\\left(\\frac{7\\pi}{2}\\right)L(x)=\u2212sin(27\u03c0\u200b)(x\u221227\u03c0\u200b)+cos(27\u03c0\u200b)<\/p>\n\n\n\n<p>To analyze and simplify this expression, let&#8217;s start by evaluating the sine and cosine functions at 7\u03c02\\frac{7\\pi}{2}27\u03c0\u200b.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Evaluate sin\u2061(7\u03c02)\\sin\\left(\\frac{7\\pi}{2}\\right)sin(27\u03c0\u200b) and cos\u2061(7\u03c02)\\cos\\left(\\frac{7\\pi}{2}\\right)cos(27\u03c0\u200b)<\/h3>\n\n\n\n<p>The angle 7\u03c02\\frac{7\\pi}{2}27\u03c0\u200b is greater than 2\u03c02\\pi2\u03c0, so it corresponds to an angle that has gone around the unit circle more than once.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>sin\u2061(7\u03c02)\\sin\\left(\\frac{7\\pi}{2}\\right)sin(27\u03c0\u200b):<br>Since 7\u03c02=2\u03c0+3\u03c02\\frac{7\\pi}{2} = 2\\pi + \\frac{3\\pi}{2}27\u03c0\u200b=2\u03c0+23\u03c0\u200b, this is effectively equivalent to the angle 3\u03c02\\frac{3\\pi}{2}23\u03c0\u200b on the unit circle. The sine of 3\u03c02\\frac{3\\pi}{2}23\u03c0\u200b is \u22121-1\u22121, so: sin\u2061(7\u03c02)=\u22121\\sin\\left(\\frac{7\\pi}{2}\\right) = -1sin(27\u03c0\u200b)=\u22121<\/li>\n\n\n\n<li>cos\u2061(7\u03c02)\\cos\\left(\\frac{7\\pi}{2}\\right)cos(27\u03c0\u200b):<br>Similarly, 7\u03c02\\frac{7\\pi}{2}27\u03c0\u200b is coterminal with 3\u03c02\\frac{3\\pi}{2}23\u03c0\u200b, so the cosine of 3\u03c02\\frac{3\\pi}{2}23\u03c0\u200b is 000. Hence: cos\u2061(7\u03c02)=0\\cos\\left(\\frac{7\\pi}{2}\\right) = 0cos(27\u03c0\u200b)=0<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Simplify the Expression<\/h3>\n\n\n\n<p>Now, substitute these values into the original expression for L(x)L(x)L(x):L(x)=\u2212(\u22121)(x\u22127\u03c02)+0L(x) = -(-1) \\left( x &#8211; \\frac{7\\pi}{2} \\right) + 0L(x)=\u2212(\u22121)(x\u221227\u03c0\u200b)+0L(x)=(x\u22127\u03c02)L(x) = \\left( x &#8211; \\frac{7\\pi}{2} \\right)L(x)=(x\u221227\u03c0\u200b)<\/p>\n\n\n\n<p>So the simplified function is:L(x)=x\u22127\u03c02L(x) = x &#8211; \\frac{7\\pi}{2}L(x)=x\u221227\u03c0\u200b<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Interpretation<\/h3>\n\n\n\n<p>The function L(x)=x\u22127\u03c02L(x) = x &#8211; \\frac{7\\pi}{2}L(x)=x\u221227\u03c0\u200b is a linear function, where the slope is 111 and the y-intercept is \u22127\u03c02-\\frac{7\\pi}{2}\u221227\u03c0\u200b. This means that for every unit increase in xxx, L(x)L(x)L(x) increases by 1.<\/p>\n\n\n\n<p>In summary:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The function simplifies to L(x)=x\u22127\u03c02L(x) = x &#8211; \\frac{7\\pi}{2}L(x)=x\u221227\u03c0\u200b.<\/li>\n\n\n\n<li>The original sine and cosine terms simplify due to the specific values of sin\u2061\\sinsin and cos\u2061\\coscos at 7\u03c02\\frac{7\\pi}{2}27\u03c0\u200b, which ultimately result in a linear equation.<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-155.jpeg\" alt=\"\" class=\"wp-image-46477\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-155.jpeg 1024w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-155-300x300.jpeg 300w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-155-150x150.jpeg 150w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-155-768x768.jpeg 768w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>L(x) = -sin(7pi\/2)(x &#8211; 7pi\/2) + cos(7pi\/2) The Correct Answer and Explanation is: The function you&#8217;ve provided is:L(x)=\u2212sin\u2061(7\u03c02)(x\u22127\u03c02)+cos\u2061(7\u03c02)L(x) = -\\sin\\left(\\frac{7\\pi}{2}\\right) \\left( x &#8211; \\frac{7\\pi}{2} \\right) + \\cos\\left(\\frac{7\\pi}{2}\\right)L(x)=\u2212sin(27\u03c0\u200b)(x\u221227\u03c0\u200b)+cos(27\u03c0\u200b) To analyze and simplify this expression, let&#8217;s start by evaluating the sine and cosine functions at 7\u03c02\\frac{7\\pi}{2}27\u03c0\u200b. Step 1: Evaluate sin\u2061(7\u03c02)\\sin\\left(\\frac{7\\pi}{2}\\right)sin(27\u03c0\u200b) and cos\u2061(7\u03c02)\\cos\\left(\\frac{7\\pi}{2}\\right)cos(27\u03c0\u200b) The angle 7\u03c02\\frac{7\\pi}{2}27\u03c0\u200b is greater [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-46473","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/46473","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=46473"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/46473\/revisions"}],"predecessor-version":[{"id":46478,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/46473\/revisions\/46478"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=46473"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=46473"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=46473"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}