Let’s solve each part of your query one at a time.<\/p>\n\n\n\n
Part A: Trigonometry Problem<\/h3>\n\n\n\n
You are asked to expand the expression (\\sin \\left(\\frac{3\\pi}{2} – \\frac{3\\pi}{4}\\right)) using the angle subtraction formula for sine, which is:<\/p>\n\n\n\n
[
\\sin(a – b) = \\sin a \\cos b – \\cos a \\sin b
]<\/p>\n\n\n\n
Plugging in the values (a = \\frac{3\\pi}{2}) and (b = \\frac{3\\pi}{4}), we have:<\/p>\n\n\n\n
[
\\sin \\left(\\frac{3\\pi}{2} – \\frac{3\\pi}{4}\\right) = \\sin \\frac{3\\pi}{2} \\cos \\frac{3\\pi}{4} – \\cos \\frac{3\\pi}{2} \\sin \\frac{3\\pi}{4}
]<\/p>\n\n\n\n
To evaluate this, remember the values of sine and cosine at these specific angles:<\/p>\n\n\n\n
- \n
- (\\sin \\frac{3\\pi}{2} = -1)<\/li>\n\n\n\n
- (\\cos \\frac{3\\pi}{2} = 0)<\/li>\n\n\n\n
- (\\sin \\frac{3\\pi}{4} = \\frac{\\sqrt{2}}{2})<\/li>\n\n\n\n
- (\\cos \\frac{3\\pi}{4} = -\\frac{\\sqrt{2}}{2})<\/li>\n<\/ul>\n\n\n\n
Substituting these into the formula gives:<\/p>\n\n\n\n
[
(-1)(-\\frac{\\sqrt{2}}{2}) – (0)(\\frac{\\sqrt{2}}{2}) = \\frac{\\sqrt{2}}{2}
]<\/p>\n\n\n\nThis matches the expansion:
[
\\sin \\frac{3\\pi}{2} \\cos \\frac{3\\pi}{4} + \\cos \\frac{3\\pi}{2} \\sin \\frac{3\\pi}{4}
]<\/p>\n\n\n\nSo, the correct option is D.<\/strong><\/p>\n\n\n\n
Part B: Algebra Problem<\/h3>\n\n\n\n
The expression for subtracting (4x + 6) from (7 + 2x) is calculated as follows:<\/p>\n\n\n\n
[
(7 + 2x) – (4x + 6)
]<\/p>\n\n\n\nDistribute the negative sign through the second term:
[
7 + 2x – 4x – 6
]<\/p>\n\n\n\nCombine like terms:
[
1 – 2x
]<\/p>\n\n\n\nThis represents the result of the subtraction: “1 – 2x.” This result is simply obtained by direct subtraction and combining like terms. The subtraction involves distributing the minus sign across (4x + 6) and then consolidating the x terms and the constant terms independently.<\/p>\n\n\n\n
Generating the Image<\/h3>\n\n\n\n
Now let’s create an image that visually represents the algebraic expression subtraction process:<\/p>\n\n\n\n
- \n
- Image Description:<\/strong> A chalkboard with the algebraic expression (7 + 2x – (4x + 6)) written in white chalk, showing each step of the subtraction: distribution of the negative sign, combining like terms, and resulting in (1 – 2x). Include visual arrows and annotations explaining each step.<\/li>\n<\/ul>\n\n\n\n
I’ll generate this image now.<\/p>\n\n\n\n
Here’s the image representing the step-by-step subtraction of the algebraic expression (7 + 2x – (4x + 6)). Each step is clearly annotated to show how the terms are combined to arrive at the final expression (1 – 2x). You can view the visual explanation on the chalkboard in the image above.<\/p>\n\n\n\n
![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/02\/image-1132.png\")
<\/figure>\n","protected":false},"excerpt":{"rendered":"A. sin (3pi\/2 – 3pi\/4) can be expanded to: A. cos 3pi\/2 cos 3pi\/4 + sin 3pi\/2 cos 3pi\/4 B. sin 3pi\/2 cos 3pi\/4 – cos 3pi\/2 sin 3pi\/4 C. cos 3pi\/2 cos 3pi\/4 – sin 3pi\/2 sin 3pi\/4 D. sin 3pi\/2 cos 3pi\/4 + cos 3pi\/2 sin 3pi\/4 B. Thick expression is obtained on […]<\/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-193805","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/193805","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=193805"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/193805\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=193805"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=193805"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=193805"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
- Image Description:<\/strong> A chalkboard with the algebraic expression (7 + 2x – (4x + 6)) written in white chalk, showing each step of the subtraction: distribution of the negative sign, combining like terms, and resulting in (1 – 2x). Include visual arrows and annotations explaining each step.<\/li>\n<\/ul>\n\n\n\n