find all solutions t between 360 and 720 degrees, inclusive: <\/p>\n\n\n\n
(a) cos t = sin t <\/p>\n\n\n\n
(b) ta t = \u20134.3315 <\/p>\n\n\n\n
(c) sin t = \u20130.9397<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n Let’s solve each part step by step.<\/p>\n\n\n\n Each trigonometric equation can be solved by recognizing the relevant identities and using inverse trigonometric functions. In (a), ( \\cos t = \\sin t ) simplifies to ( \\tan t = 1 ), and since tangent has a period of 180\u00b0, solutions are found by adding multiples of 180\u00b0 to the reference angle of 45\u00b0. Thus, the solutions between 360\u00b0 and 720\u00b0 are 405\u00b0 and 585\u00b0.<\/p>\n\n\n\n In (b), we solve ( \\tan t = -4.3315 ) by applying the inverse tangent function. The initial angle is approximately -77.1\u00b0, which can be adjusted by adding multiples of 180\u00b0 to find solutions in the specified interval. This yields solutions at 462.9\u00b0 and 642.9\u00b0.<\/p>\n\n\n\n In (c), ( \\sin t = -0.9397 ) leads to ( t = \\sin^{-1}(-0.9397) ), which gives a reference angle of approximately -70\u00b0. Since sine is negative in the third and fourth quadrants, we add 360\u00b0 and appropriate multiples of 180\u00b0 to find the final answers, resulting in solutions at 430\u00b0 and 650\u00b0.<\/p>\n\n\n\n By understanding periodicity and quadrant behavior, we can systematically find all solutions within the desired range.<\/p>\n","protected":false},"excerpt":{"rendered":" find all solutions t between 360 and 720 degrees, inclusive: (a) cos t = sin t (b) ta t = \u20134.3315 (c) sin t = \u20130.9397 The Correct Answer and Explanation is : Let’s solve each part step by step. (a) ( \\cos t = \\sin t ) (b) ( \\tan t = -4.3315 ) […]<\/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-155184","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/155184","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=155184"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/155184\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=155184"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=155184"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=155184"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}(a) ( \\cos t = \\sin t )<\/h3>\n\n\n\n
\n
[
\\cos t = \\sin t
]
Divide both sides by ( \\cos t ) (assuming ( \\cos t \\neq 0 )):
[
1 = \\tan t
]
So, we need to find where ( \\tan t = 1 ).<\/li>\n\n\n\n
The general solution for ( \\tan t = 1 ) is:
[
t = 45^\\circ + 180^\\circ k
]
where ( k ) is an integer.<\/li>\n\n\n\n
For ( k = 2 ):
[
t = 45^\\circ + 180^\\circ \\times 2 = 405^\\circ
]
For ( k = 3 ):
[
t = 45^\\circ + 180^\\circ \\times 3 = 585^\\circ
]
So the solutions for this part are:
[
t = 405^\\circ, 585^\\circ
]<\/li>\n<\/ol>\n\n\n\n
\n\n\n\n(b) ( \\tan t = -4.3315 )<\/h3>\n\n\n\n
\n
[
t = \\tan^{-1}(-4.3315)
]
Using a calculator:
[
t \\approx -77.1^\\circ
]<\/li>\n\n\n\n
Since the tangent function has a period of 180\u00b0, add 180\u00b0 repeatedly to find solutions within the given range. For the first solution, add 360\u00b0:
[
t = -77.1^\\circ + 360^\\circ \\approx 282.9^\\circ
]
Add 180\u00b0 to move it into the desired range:
[
t = 282.9^\\circ + 180^\\circ \\approx 462.9^\\circ
]
Add another 180\u00b0:
[
t = 462.9^\\circ + 180^\\circ \\approx 642.9^\\circ
]
So, the solutions are:
[
t = 462.9^\\circ, 642.9^\\circ
]<\/li>\n<\/ol>\n\n\n\n
\n\n\n\n(c) ( \\sin t = -0.9397 )<\/h3>\n\n\n\n
\n
[
t = \\sin^{-1}(-0.9397)
]
Using a calculator:
[
t \\approx -70^\\circ
]<\/li>\n\n\n\n
The sine function is negative in the third and fourth quadrants, so the general solution is:
[
t = 360^\\circ + 70^\\circ = 430^\\circ
]
and
[
t = 360^\\circ + 180^\\circ – 70^\\circ = 650^\\circ
]
So the solutions are:
[
t = 430^\\circ, 650^\\circ
]<\/li>\n<\/ol>\n\n\n\nSummary of solutions:<\/h3>\n\n\n\n
\n
Explanation (300 words):<\/h3>\n\n\n\n