{"id":7844,"date":"2023-07-28T10:51:00","date_gmt":"2023-07-28T05:21:00","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=7844"},"modified":"2023-07-28T16:43:44","modified_gmt":"2023-07-28T11:13:44","slug":"trigonometry-ratios-of-complementary-angles","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/trigonometry-ratios-of-complementary-angles\/","title":{"rendered":"Trigonometry ratios of complementary angles"},"content":{"rendered":"\n

Trigonometry is a branch of mathematics that deals with the relationship between the sides and angles of a triangle. In this article, we will discuss the trigonometry ratios of complementary angles. <\/p>\n\n\n\n

Complementary angles <\/h2>\n\n\n\n

Complementary angles are two angles whose sum is 90 degrees<\/p>\n\n\n\n

if A and B are complementary angles, then<\/p>\n\n\n\n

$\\angle A + \\angle B = 90^0$
or
$\\angle B = 90 – \\angle A$<\/p>\n\n\n\n

Hence A and 90 – A represent complementary angles<\/p>\n\n\n\n

Complementary angles in Right angle triangle<\/h2>\n\n\n\n
\"Trigonometry<\/a><\/figure>\n\n\n\n

In a right angle triangle ABC
A+B+C=180
Now B=90
So A+C=90
Or 
A=90?C<\/p>\n\n\n\n

Hence the acute angles in right angle triangle are complementary<\/p>\n\n\n\n

Definition of Trigonometry Ratios of Complementary Angles<\/h2>\n\n\n\n

We know that trigonometric ratios of the acute angles A and C are given by<\/p>\n\n\n\n

Angle C<\/strong><\/p>\n\n\n\n

$\\sin C= \\frac {\\text{Perpendicular}}{\\text{Hypotenuse}}=\\frac {AB}{AC}$

$\\csc C= \\frac {\\text{Hypotenuse}}{\\text{Perpendicular}} =\\frac {AC}{AB}$

$\\cos C= \\frac {\\text{Base}}{\\text{Hypotenuse}}=\\frac {BC}{AC}$

$\\sec C= \\frac {\\text{Hypotenuse}}{\\text{Base}}=\\frac {AC}{BC}$

$\\tan C= \\frac {\\text{Perpendicular}}{\\text{Base}}=\\frac {AB}{BC}$

$\\cot C= \\frac {\\text{Base}}{Perpendicular}=\\frac {BC}{AB}$<\/p>\n\n\n\n

Angle A<\/strong><\/p>\n\n\n\n

$\\sin A= \\frac {\\text{Perpendicular}}{\\text{Hypotenuse}}=\\frac {BC}{AC}$

$\\csc A= \\frac {\\text{Hypotenuse}}{\\text{Perpendicular}} =\\frac {AC}{BC}$

$\\cos A= \\frac {\\text{Base}}{\\text{Hypotenuse}}=\\frac {AB}{AC}$

$\\sec A = \\frac {\\text{Hypotenuse}}{\\text{Base}}=\\frac {AC}{AB}$

$\\tan A= \\frac {\\text{Perpendicular}}{\\text{Base}}=\\frac {BC}{AB}$

$\\cot A = \\frac {\\text{Base}}{Perpendicular}=\\frac {AB}{BC}$<\/p>\n\n\n\n

Now in a right angle triangle <\/p>\n\n\n\n

C =90 -A<\/p>\n\n\n\n

Therefore from Angle C ration’s will become<\/p>\n\n\n\n

$\\sin (90 -A)=\\frac {AB}{AC}$

$\\csc (90 -A) =\\frac {AC}{AB}$

$\\cos (90 -A) =\\frac {BC}{AC}$

$\\sec (90 -A) =\\frac {AC}{BC}$

$\\tan (90 -A) =\\frac {AB}{BC}$

$\\cot(90 -A) =\\frac {BC}{AB}$<\/p>\n\n\n\n

From these two equations we can find that<\/p>\n\n\n\n

sin(A) = cos(90\u00b0 – A)
cos(A) = sin(90\u00b0 – A)
tan(A) = cot(90\u00b0 – A)
cot(A) = tan(90\u00b0 – A)
sec(A) = csc(90\u00b0 – A)
csc(A) = sec(90\u00b0 – A)<\/p>\n\n\n\n

The above is the called the trigonometric ratio’s of the complementary angles<\/p>\n\n\n\n

Examples of Trigonometry Ratios of Complementary Angles<\/h2>\n\n\n\n

Let us take an example of a right triangle with an acute angle A measuring 30 degrees. The complementary angle B will measure 60 degrees.<\/p>\n\n\n\n

Using the formulas, we can find the trigonometry ratios of angle A and angle B.<\/p>\n\n\n\n

sin(30\u00b0) = cos(60\u00b0) = 1\/2
cos(30\u00b0) = sin(60\u00b0) = $\\frac {\\sqrt 3}{2}$
tan(30\u00b0) = cot(60\u00b0) = $\\frac {1}{\\sqrt 3}$
cot(30\u00b0) = tan(60\u00b0) = $\\sqrt 3$
sec(30\u00b0) = csc(60\u00b0) = $\\frac {2}{\\sqrt 3}$
csc(30\u00b0) = sec(60\u00b0) = 2<\/p>\n\n\n\n

Similarly, for an acute angle A measuring 45 degrees, the complementary angle B will measure 45 degrees as well.<\/p>\n\n\n\n

sin(45\u00b0) = cos(45\u00b0) =$\\frac { 1}{\\sqrt 2}$
tan(45\u00b0) = cot(45\u00b0) = 1
sec(45\u00b0) = csc(45\u00b0) = $\\sqrt 2$<\/p>\n\n\n\n

Sample Questions<\/h2>\n\n\n\n
    \n
  1. If sin(A) = 3\/5, find cos(A) and tan(90\u00b0 – A).<\/li>\n<\/ol>\n\n\n\n

    Solution:
    Given, sin(A) = 3\/5
    $cos(A) = \\sqrt {(1 – sin^2(A))} = \\sqrt {(1 – (3\/5)^2)} = 4\/5$
    tan(90\u00b0 – A) = cot(A) = cos(A)\/sin(A) = (4\/5)\/(3\/5) = 4\/3<\/p>\n\n\n\n

      \n
    1. If tan(A) = 7\/24, find sin(90\u00b0 – A) and csc(A).<\/li>\n<\/ol>\n\n\n\n

      Solution:
      Given, tan(A) = 7\/24
      $sin(A) = tan(A)\/\\sqrt {(1 + tan^2(A))} = 7\/25$
      $sin(90\u00b0 – A) = cos(A) = \\sqrt {(1 – sin^2(A))} =24\/25$
      csc(A) =1\/sin (A) =25\/7<\/p>\n\n\n\n

      I hope you like this article on Trigonometry ratios of complementary angles<\/p>\n\n\n\n

      Related Articles<\/strong><\/p>\n\n\n\n

      Trigonometric ratios of angle A in terms of sin A<\/a>
      Reciprocal of cos function<\/a>
      Trigonometry formula for triangle<\/a>
      <\/p>\n","protected":false},"excerpt":{"rendered":"

      Trigonometry is a branch of mathematics that deals with the relationship between the sides and angles of a triangle. In this article, we will discuss the trigonometry ratios of complementary angles. Complementary angles Complementary angles are two angles whose sum is 90 degrees if A and B are complementary angles, then $\\angle A + \\angle […]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","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":"","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":[498],"tags":[],"class_list":["post-7844","post","type-post","status-publish","format-standard","hentry","category-maths"],"yoast_head":"\nTrigonometry ratios of complementary angles - 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In this article, we will discuss the trigonometry ratios of complementary angles. Complementary angles Complementary angles are two angles whose sum is 90 degrees if A and B are complementary angles, then $\\angle A + \\angle…","_links":{"self":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/7844","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/users\/8"}],"replies":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/comments?post=7844"}],"version-history":[{"count":3,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/7844\/revisions"}],"predecessor-version":[{"id":8189,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/7844\/revisions\/8189"}],"wp:attachment":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/media?parent=7844"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/categories?post=7844"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/tags?post=7844"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}