To evaluate the expression: sin\u206170\u2218+cos\u206140\u2218bycos\u206170\u2218+sin\u206140\u2218\\sin 70^\\circ + \\cos 40^\\circ \\quad \\text{by} \\quad \\cos 70^\\circ + \\sin 40^\\circsin70\u2218+cos40\u2218bycos70\u2218+sin40\u2218<\/p>\n\n\n\n
First, recall that: cos\u2061(90\u2218\u2212x)=sin\u2061(x)andsin\u2061(90\u2218\u2212x)=cos\u2061(x)\\cos(90^\\circ – x) = \\sin(x) \\quad \\text{and} \\quad \\sin(90^\\circ – x) = \\cos(x)cos(90\u2218\u2212x)=sin(x)andsin(90\u2218\u2212x)=cos(x)<\/p>\n\n\n\n
Step 1: Express terms using co-function identities<\/h3>\n\n\n\n
We know that:<\/p>\n\n\n\n
- \n
- cos\u206140\u2218=sin\u2061(90\u2218\u221240\u2218)=sin\u206150\u2218\\cos 40^\\circ = \\sin (90^\\circ – 40^\\circ) = \\sin 50^\\circcos40\u2218=sin(90\u2218\u221240\u2218)=sin50\u2218<\/li>\n\n\n\n
- sin\u206140\u2218=cos\u2061(90\u2218\u221240\u2218)=cos\u206150\u2218\\sin 40^\\circ = \\cos (90^\\circ – 40^\\circ) = \\cos 50^\\circsin40\u2218=cos(90\u2218\u221240\u2218)=cos50\u2218<\/li>\n<\/ul>\n\n\n\n
Now, we can rewrite the expression as: sin\u206170\u2218+cos\u206140\u2218becomessin\u206170\u2218+sin\u206150\u2218\\sin 70^\\circ + \\cos 40^\\circ \\quad \\text{becomes} \\quad \\sin 70^\\circ + \\sin 50^\\circsin70\u2218+cos40\u2218becomessin70\u2218+sin50\u2218 cos\u206170\u2218+sin\u206140\u2218becomescos\u206170\u2218+cos\u206150\u2218\\cos 70^\\circ + \\sin 40^\\circ \\quad \\text{becomes} \\quad \\cos 70^\\circ + \\cos 50^\\circcos70\u2218+sin40\u2218becomescos70\u2218+cos50\u2218<\/p>\n\n\n\n
Step 2: Simplify using sum of angles formula<\/h3>\n\n\n\n
We can use the sine and cosine addition formulas for further simplification.<\/p>\n\n\n\n
- \n
- The sum of sine terms:<\/li>\n<\/ul>\n\n\n\n
sin\u2061A+sin\u2061B=2sin\u2061(A+B2)cos\u2061(A\u2212B2)\\sin A + \\sin B = 2 \\sin\\left(\\frac{A + B}{2}\\right) \\cos\\left(\\frac{A – B}{2}\\right)sinA+sinB=2sin(2A+B\u200b)cos(2A\u2212B\u200b)<\/p>\n\n\n\n
- \n
- The sum of cosine terms:<\/li>\n<\/ul>\n\n\n\n
cos\u2061A+cos\u2061B=2cos\u2061(A+B2)cos\u2061(A\u2212B2)\\cos A + \\cos B = 2 \\cos\\left(\\frac{A + B}{2}\\right) \\cos\\left(\\frac{A – B}{2}\\right)cosA+cosB=2cos(2A+B\u200b)cos(2A\u2212B\u200b)<\/p>\n\n\n\n
For sin\u206170\u2218+sin\u206150\u2218\\sin 70^\\circ + \\sin 50^\\circsin70\u2218+sin50\u2218, applying the sine addition formula: sin\u206170\u2218+sin\u206150\u2218=2sin\u2061(70\u2218+50\u22182)cos\u2061(70\u2218\u221250\u22182)\\sin 70^\\circ + \\sin 50^\\circ = 2 \\sin\\left(\\frac{70^\\circ + 50^\\circ}{2}\\right) \\cos\\left(\\frac{70^\\circ – 50^\\circ}{2}\\right)sin70\u2218+sin50\u2218=2sin(270\u2218+50\u2218\u200b)cos(270\u2218\u221250\u2218\u200b) =2sin\u206160\u2218cos\u206110\u2218= 2 \\sin 60^\\circ \\cos 10^\\circ=2sin60\u2218cos10\u2218<\/p>\n\n\n\n
For cos\u206170\u2218+cos\u206150\u2218\\cos 70^\\circ + \\cos 50^\\circcos70\u2218+cos50\u2218, applying the cosine addition formula: cos\u206170\u2218+cos\u206150\u2218=2cos\u2061(70\u2218+50\u22182)cos\u2061(70\u2218\u221250\u22182)\\cos 70^\\circ + \\cos 50^\\circ = 2 \\cos\\left(\\frac{70^\\circ + 50^\\circ}{2}\\right) \\cos\\left(\\frac{70^\\circ – 50^\\circ}{2}\\right)cos70\u2218+cos50\u2218=2cos(270\u2218+50\u2218\u200b)cos(270\u2218\u221250\u2218\u200b) =2cos\u206160\u2218cos\u206110\u2218= 2 \\cos 60^\\circ \\cos 10^\\circ=2cos60\u2218cos10\u2218<\/p>\n\n\n\n
Step 3: Final simplification<\/h3>\n\n\n\n
We know: sin\u206160\u2218=32andcos\u206160\u2218=12\\sin 60^\\circ = \\frac{\\sqrt{3}}{2} \\quad \\text{and} \\quad \\cos 60^\\circ = \\frac{1}{2}sin60\u2218=23\u200b\u200bandcos60\u2218=21\u200b<\/p>\n\n\n\n
Thus, the expression becomes: 2sin\u206160\u2218cos\u206110\u2218=2\u00d732\u00d7cos\u206110\u2218=3cos\u206110\u22182 \\sin 60^\\circ \\cos 10^\\circ = 2 \\times \\frac{\\sqrt{3}}{2} \\times \\cos 10^\\circ = \\sqrt{3} \\cos 10^\\circ2sin60\u2218cos10\u2218=2\u00d723\u200b\u200b\u00d7cos10\u2218=3\u200bcos10\u2218 2cos\u206160\u2218cos\u206110\u2218=2\u00d712\u00d7cos\u206110\u2218=cos\u206110\u22182 \\cos 60^\\circ \\cos 10^\\circ = 2 \\times \\frac{1}{2} \\times \\cos 10^\\circ = \\cos 10^\\circ2cos60\u2218cos10\u2218=2\u00d721\u200b\u00d7cos10\u2218=cos10\u2218<\/p>\n\n\n\n
Step 4: Final result<\/h3>\n\n\n\n
So, the original expression simplifies to: 3cos\u206110\u2218cos\u206110\u2218=3\\frac{\\sqrt{3} \\cos 10^\\circ}{\\cos 10^\\circ} = \\sqrt{3}cos10\u22183\u200bcos10\u2218\u200b=3\u200b<\/p>\n\n\n\n
Thus, the value of sin\u206170\u2218+cos\u206140\u2218\\sin 70^\\circ + \\cos 40^\\circsin70\u2218+cos40\u2218 by cos\u206170\u2218+sin\u206140\u2218\\cos 70^\\circ + \\sin 40^\\circcos70\u2218+sin40\u2218 is approximately: 3\\boxed{\\sqrt{3}}3\u200b\u200b<\/p>\n\n\n\n
![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner6-1408.jpeg\")
<\/figure>\n","protected":false},"excerpt":{"rendered":"The value of sin 70 degrees + cos 40 Degrees by cos 70 degrees plus sin 40 Degrees The Correct Answer and Explanation is: To evaluate the expression: sin\u206170\u2218+cos\u206140\u2218bycos\u206170\u2218+sin\u206140\u2218\\sin 70^\\circ + \\cos 40^\\circ \\quad \\text{by} \\quad \\cos 70^\\circ + \\sin 40^\\circsin70\u2218+cos40\u2218bycos70\u2218+sin40\u2218 First, recall that: cos\u2061(90\u2218\u2212x)=sin\u2061(x)andsin\u2061(90\u2218\u2212x)=cos\u2061(x)\\cos(90^\\circ – x) = \\sin(x) \\quad \\text{and} \\quad \\sin(90^\\circ – x) […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-264763","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/264763","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=264763"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/264763\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=264763"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=264763"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=264763"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
- The sum of cosine terms:<\/li>\n<\/ul>\n\n\n\n
- The sum of sine terms:<\/li>\n<\/ul>\n\n\n\n