{"id":34871,"date":"2025-06-23T20:06:47","date_gmt":"2025-06-23T20:06:47","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=34871"},"modified":"2025-06-23T20:06:49","modified_gmt":"2025-06-23T20:06:49","slug":"find-the-exact-value-of-the-following-expression","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/find-the-exact-value-of-the-following-expression\/","title":{"rendered":"Find the exact value of the following expression."},"content":{"rendered":"\n<p>Find the exact value of the following expression. \\tan\\left(\\frac{\\pi}{6} + \\frac{5\\pi}{4}\\right) Rewrite the expression using a sum or difference formula. Choose the correct answer below. \\frac{\\tan\\frac{5\\pi}{4} &#8211; \\tan\\frac{\\pi}{6}}{1 &#8211; \\tan\\frac{\\pi}{6}\\tan\\frac{5\\pi}{4}} A. \\frac{\\tan\\frac{\\pi}{6} + \\tan\\frac{5\\pi}{4}}{1 &#8211; \\tan\\frac{\\pi}{6}\\tan\\frac{5\\pi}{4}} C. \\frac{\\sin\\frac{\\pi}{6}\\cos\\frac{5\\pi}{4} + \\cos\\frac{\\pi}{6}\\sin\\frac{5\\pi}{4}}{} B. \\frac{\\tan\\frac{\\pi}{6} &#8211; \\tan\\frac{5\\pi}{4}}{1 + \\tan\\frac{\\pi}{6}\\tan\\frac{5\\pi}{4}} D. Find the exact value of the expression. \\tan\\left(\\frac{\\pi}{6} + \\frac{5\\pi}{4}\\right) = \\sqrt{7 + 4\\sqrt{3}} (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression. Rationalize all denominators.)<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"530\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-265.png\" alt=\"\" class=\"wp-image-34872\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-265.png 1024w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-265-300x155.png 300w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-265-768x398.png 768w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-green-cyan-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To evaluate<\/p>\n\n\n\n<p>tan\u2061(\u03c06+5\u03c04)\\tan\\left(\\frac{\\pi}{6} + \\frac{5\\pi}{4}\\right)<\/p>\n\n\n\n<p>we use the tangent sum identity:<\/p>\n\n\n\n<p>tan\u2061(A+B)=tan\u2061A+tan\u2061B1\u2212tan\u2061Atan\u2061B\\tan(A + B) = \\frac{\\tan A + \\tan B}{1 &#8211; \\tan A \\tan B}<\/p>\n\n\n\n<p>Let A=\u03c06A = \\frac{\\pi}{6} and B=5\u03c04B = \\frac{5\\pi}{4}. Applying the formula:<\/p>\n\n\n\n<p>tan\u2061(\u03c06+5\u03c04)=tan\u2061\u03c06+tan\u20615\u03c041\u2212tan\u2061\u03c06tan\u20615\u03c04\\tan\\left(\\frac{\\pi}{6} + \\frac{5\\pi}{4}\\right) = \\frac{\\tan\\frac{\\pi}{6} + \\tan\\frac{5\\pi}{4}}{1 &#8211; \\tan\\frac{\\pi}{6} \\tan\\frac{5\\pi}{4}}<\/p>\n\n\n\n<p>From the unit circle:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>tan\u2061\u03c06=13\\tan\\frac{\\pi}{6} = \\frac{1}{\\sqrt{3}}<\/li>\n\n\n\n<li>tan\u20615\u03c04=1\\tan\\frac{5\\pi}{4} = 1<\/li>\n<\/ul>\n\n\n\n<p>Substituting:<\/p>\n\n\n\n<p>=13+11\u2212(13\u22c51)=1+33\u22121= \\frac{\\frac{1}{\\sqrt{3}} + 1}{1 &#8211; \\left(\\frac{1}{\\sqrt{3}} \\cdot 1\\right)} = \\frac{1 + \\sqrt{3}}{\\sqrt{3} &#8211; 1}<\/p>\n\n\n\n<p>Rationalizing the denominator:<\/p>\n\n\n\n<p>1+33\u22121\u22c53+13+1=(1+3)(3+1)(3)2\u221212\\frac{1 + \\sqrt{3}}{\\sqrt{3} &#8211; 1} \\cdot \\frac{\\sqrt{3} + 1}{\\sqrt{3} + 1} = \\frac{(1 + \\sqrt{3})(\\sqrt{3} + 1)}{(\\sqrt{3})^2 &#8211; 1^2}<\/p>\n\n\n\n<p>=(1\u22c53+1\u22c51+3\u22c53+3\u22c51)3\u22121=3+1+3+32=4+232=2+3= \\frac{(1 \\cdot \\sqrt{3} + 1 \\cdot 1 + \\sqrt{3} \\cdot \\sqrt{3} + \\sqrt{3} \\cdot 1)}{3 &#8211; 1} = \\frac{\\sqrt{3} + 1 + 3 + \\sqrt{3}}{2} = \\frac{4 + 2\\sqrt{3}}{2} = 2 + \\sqrt{3}<\/p>\n\n\n\n<p>Hence, the exact value is 2+3\\boxed{2 + \\sqrt{3}}.<\/p>\n\n\n\n<p>This confirms that the correct choice for rewriting the expression is:<\/p>\n\n\n\n<p>tan\u2061\u03c06+tan\u20615\u03c041\u2212tan\u2061\u03c06tan\u20615\u03c04\\frac{\\tan\\frac{\\pi}{6} + \\tan\\frac{5\\pi}{4}}{1 &#8211; \\tan\\frac{\\pi}{6} \\tan\\frac{5\\pi}{4}}<\/p>\n\n\n\n<p>which corresponds to <strong>option A<\/strong>.<\/p>\n\n\n\n<p>The final numerical expression 2+32 + \\sqrt{3} is not equal to 7+43\\sqrt{7 + 4\\sqrt{3}}, despite appearances. Squaring both:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(2+3)2=4+43+3=7+43(2 + \\sqrt{3})^2 = 4 + 4\\sqrt{3} + 3 = 7 + 4\\sqrt{3}<\/li>\n<\/ul>\n\n\n\n<p>Thus, 2+3=7+432 + \\sqrt{3} = \\sqrt{7 + 4\\sqrt{3}}, confirming the equality.<\/p>\n\n\n\n<p>In summary:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Use the tangent sum formula<\/li>\n\n\n\n<li>Plug in known tangent values<\/li>\n\n\n\n<li>Rationalize carefully<\/li>\n\n\n\n<li>Confirm numerical identity by squaring<\/li>\n<\/ul>\n\n\n\n<p>Beautifully satisfying, isn&#8217;t it?<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"852\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-599.jpeg\" alt=\"\" class=\"wp-image-34877\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-599.jpeg 852w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-599-250x300.jpeg 250w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-599-768x923.jpeg 768w\" sizes=\"auto, (max-width: 852px) 100vw, 852px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Find the exact value of the following expression. \\tan\\left(\\frac{\\pi}{6} + \\frac{5\\pi}{4}\\right) Rewrite the expression using a sum or difference formula. Choose the correct answer below. \\frac{\\tan\\frac{5\\pi}{4} &#8211; \\tan\\frac{\\pi}{6}}{1 &#8211; \\tan\\frac{\\pi}{6}\\tan\\frac{5\\pi}{4}} A. \\frac{\\tan\\frac{\\pi}{6} + \\tan\\frac{5\\pi}{4}}{1 &#8211; \\tan\\frac{\\pi}{6}\\tan\\frac{5\\pi}{4}} C. \\frac{\\sin\\frac{\\pi}{6}\\cos\\frac{5\\pi}{4} + \\cos\\frac{\\pi}{6}\\sin\\frac{5\\pi}{4}}{} B. \\frac{\\tan\\frac{\\pi}{6} &#8211; \\tan\\frac{5\\pi}{4}}{1 + \\tan\\frac{\\pi}{6}\\tan\\frac{5\\pi}{4}} D. Find the exact value of the expression. \\tan\\left(\\frac{\\pi}{6} [&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-34871","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/34871","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=34871"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/34871\/revisions"}],"predecessor-version":[{"id":34886,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/34871\/revisions\/34886"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=34871"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=34871"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=34871"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}