{"id":825729,"date":"2024-05-08T18:58:11","date_gmt":"2024-05-08T13:28:11","guid":{"rendered":"https:\/\/leverageedu.com\/discover\/?p=825729"},"modified":"2024-05-08T18:58:11","modified_gmt":"2024-05-08T13:28:11","slug":"basic-concept-trigonometry-formulas","status":"publish","type":"post","link":"https:\/\/leverageedu.com\/discover\/school-education\/basic-concept-trigonometry-formulas\/","title":{"rendered":"Trigonometry Formulas, Examples and Solutions!\u00a0"},"content":{"rendered":"\n<p>Trigonometry is the study of relationships between angles and sides of triangles and is a fundamental branch of mathematics. However, Trigonometry Formulas come into use whilst solving trigonometry problems. Furthermore, these Trigonometry Formulas help with the solutions. Read on to learn more about these important Trigonometry Formulas along with their Examples and Solutions.\u00a0<\/p>\n\n\n\n<p class=\"has-very-light-gray-to-cyan-bluish-gray-gradient-background has-background\"><strong>Also Read: <\/strong><a href=\"https:\/\/leverageedu.com\/blog\/wp-content\/uploads\/2021\/10\/Introduction-To-Trigonometry-Class-10-NCERT-PDF.pdf\"><strong>Introduction to Trigonometry<\/strong><\/a><\/p>\n\n\n\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-basic-trigonometric-ratio-formulas-nbsp\">Basic Trigonometric Ratio Formulas\u00a0<\/h2>\n\n\n\n<p>The Basic Trigonometric Ratios are sine (sin), cosine (cos), and tangent (tan). Moreover, these Ratios are defined as follows:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Sine (sin): The ratio of the opposite side to the hypotenuse of a right-angled triangle.\n<ul class=\"wp-block-list\">\n<li><strong>For Example:<\/strong> In a right-angled triangle with an angle of 30\u00b0, if the hypotenuse is 10 units, the opposite side is 5 units.\u00a0<\/li>\n\n\n\n<li>Therefore, sin(30\u00b0) = 0.5.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Cosine (cos): The ratio of the adjacent side to the hypotenuse of a right-angled triangle.\n<ul class=\"wp-block-list\">\n<li><strong>For Example:<\/strong> In the same triangle, the adjacent side is 8.66 units.\u00a0<\/li>\n\n\n\n<li>Therefore, cos(30\u00b0) = 0.866.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Tangent (tan): The ratio of the opposite side to the adjacent side of a right-angled triangle.\n<ul class=\"wp-block-list\">\n<li><strong>For Example:<\/strong> In the same triangle, the tangent of 30\u00b0 is 0.577, as the opposite side is 5 units and the adjacent side is 8.66 units.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p class=\"has-very-light-gray-to-cyan-bluish-gray-gradient-background has-background\"><strong>Also Read: <\/strong><a href=\"https:\/\/leverageedu.com\/discover\/indian-exams\/exam-prep-trignometry-formulas\/\"><strong>21 Trignometry Formulas for Competitive Exams\u00a0<\/strong><\/a><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-reciprocal-identities-nbsp\">Reciprocal Identities\u00a0<\/h2>\n\n\n\n<p>Additionally, the Reciprocal Identities are:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Cosecant (csc): The reciprocal of sine, csc(x) = 1\/sin(x).<\/li>\n<\/ol>\n\n\n\n<ol class=\"wp-block-list\" start=\"2\">\n<li>Secant (sec): The reciprocal of cosine, sec(x) = 1\/cos(x).<\/li>\n<\/ol>\n\n\n\n<ol class=\"wp-block-list\" start=\"3\">\n<li>Cotangent (cot): The reciprocal of tangent, cot(x) = 1\/tan(x).<\/li>\n<\/ol>\n\n\n\n<p><strong>For Example:<\/strong>\u00a0<\/p>\n\n\n\n<p>If sin(x) = 0.5,\u00a0<\/p>\n\n\n\n<p>then csc(x) = 1\/0.5 = 2.<\/p>\n\n\n\n<p class=\"has-very-light-gray-to-cyan-bluish-gray-gradient-background has-background\"><strong>Also Read: <\/strong><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/basic-concepts-algebraic-identities\/\"><strong>Algebraic Identities: Examples and Chart<\/strong><\/a><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-trigonometric-ratio-table\">Trigonometric Ratio Table<\/h2>\n\n\n\n<p>Here is the Trigonometric Ratio Table that provides the values of sine, cosine, and tangent for common angles:<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><tbody><tr><td colspan=\"4\"><strong>Trigonometric Ratio Table<\/strong><\/td><\/tr><tr><td><strong>Angle<\/strong><\/td><td><strong>Sine<\/strong><\/td><td><strong>Cosine<\/strong><\/td><td><strong>Tangent<\/strong><\/td><\/tr><tr><td>0\u00b0<\/td><td>0<\/td><td>1<\/td><td>0<\/td><\/tr><tr><td>30\u00b0<\/td><td>0.5<\/td><td>0.866<\/td><td>0.577<\/td><\/tr><tr><td>45\u00b0<\/td><td>0.707<\/td><td>0.707<\/td><td>1<\/td><\/tr><tr><td>60\u00b0<\/td><td>0.866<\/td><td>0.5<\/td><td>1.732<\/td><\/tr><tr><td>90\u00b0<\/td><td>1<\/td><td>0<\/td><td>undefined<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p><strong>For Example:<\/strong> If the angle is 45\u00b0, the sine is 0.707, the cosine is 0.707, and the tangent is 1.<\/p>\n\n\n\n<p class=\"has-very-light-gray-to-cyan-bluish-gray-gradient-background has-background\"><strong>Also Read: <\/strong><strong>Trigonometry Formulas for Quantitative Section<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-periodic-identities-nbsp\">Periodic Identities\u00a0<\/h2>\n\n\n\n<p>Moreover, the Periodic Identities describe the repeating patterns of Trigonometric functions:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>sin(x + 2\u03c0) = sin(x)<\/li>\n<\/ol>\n\n\n\n<ol class=\"wp-block-list\" start=\"2\">\n<li>cos(x + 2\u03c0) = cos(x)<\/li>\n<\/ol>\n\n\n\n<ol class=\"wp-block-list\" start=\"3\">\n<li>tan(x + \u03c0) = -tan(x)<\/li>\n<\/ol>\n\n\n\n<p><strong>For Example:<\/strong>\u00a0<\/p>\n\n\n\n<p>If sin(x) = 0.5,\u00a0<\/p>\n\n\n\n<p>then sin(x + 2\u03c0) = 0.5.<\/p>\n\n\n\n<p class=\"has-very-light-gray-to-cyan-bluish-gray-gradient-background has-background\"><strong>Also Read: <\/strong><strong>Order of Operations and PEMDAS Rule<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-co-function-identities-nbsp\">Co-Function Identities\u00a0<\/h2>\n\n\n\n<p>The Co-Function Identities relate the Trigonometric functions of Complementary angles (angles that add up to 90\u00b0):<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>sin(x) = cos(90\u00b0 \u2013 x)<\/li>\n<\/ol>\n\n\n\n<ol class=\"wp-block-list\" start=\"2\">\n<li>cos(x) = sin(90\u00b0 \u2013 x)<\/li>\n<\/ol>\n\n\n\n<ol class=\"wp-block-list\" start=\"3\">\n<li>tan(x) = cot(90\u00b0 \u2013 x)<\/li>\n<\/ol>\n\n\n\n<p><strong>For Example:<\/strong>\u00a0<\/p>\n\n\n\n<p>If sin(30\u00b0) = 0.5,\u00a0<\/p>\n\n\n\n<p>then cos(60\u00b0) = 0.5.<\/p>\n\n\n\n<p class=\"has-very-light-gray-to-cyan-bluish-gray-gradient-background has-background\"><strong>Also Read: <\/strong><strong>Conic Sections<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-sum-and-difference-identities-nbsp\">Sum and Difference Identities\u00a0<\/h2>\n\n\n\n<p>The Sum and Difference Identities describe the relationships between Trigonometric functions of the sum or difference of two angles:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>sin(A + B) = sin(A)cos(B) + cos(A)sin(B)<\/li>\n<\/ol>\n\n\n\n<ol class=\"wp-block-list\" start=\"2\">\n<li>sin(A \u2013 B) = sin(A)cos(B) \u2013 cos(A)sin(B)<\/li>\n<\/ol>\n\n\n\n<ol class=\"wp-block-list\" start=\"3\">\n<li>cos(A + B) = cos(A)cos(B) \u2013 sin(A)sin(B)<\/li>\n<\/ol>\n\n\n\n<ol class=\"wp-block-list\" start=\"4\">\n<li>cos(A \u2013 B) = cos(A)cos(B) + sin(A)sin(B)<\/li>\n<\/ol>\n\n\n\n<p><strong>For Example:<\/strong>\u00a0<\/p>\n\n\n\n<p>If sin(30\u00b0) = 0.5 and cos(45\u00b0) = 0.707,\u00a0<\/p>\n\n\n\n<p>then sin(30\u00b0 + 45\u00b0) = 0.5 \u2715 0.707 + 0.866 \u2715 0.707 = 0.866.<\/p>\n\n\n\n<p class=\"has-very-light-gray-to-cyan-bluish-gray-gradient-background has-background\"><strong>Also Read: <\/strong><strong>What is the Difference Between Degrees and Radians?<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-half-double-and-triple-identities-nbsp\">Half, Double and Triple Identities\u00a0<\/h2>\n\n\n\n<p>These Identities relate the Trigonometric functions of an angle to the functions of Half, Double, or Triple that angle:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>sin(2x) = 2sin(x)cos(x)<\/li>\n<\/ol>\n\n\n\n<ol class=\"wp-block-list\" start=\"2\">\n<li>cos(2x) = cos\u00b2(x) \u2013 sin\u00b2(x)<\/li>\n<\/ol>\n\n\n\n<ol class=\"wp-block-list\" start=\"3\">\n<li>tan(2x) = 2tan(x) \/ (1 \u2013 tan\u00b2(x))<\/li>\n<\/ol>\n\n\n\n<p><strong>For Example:<\/strong>\u00a0<\/p>\n\n\n\n<p>If sin(30\u00b0) = 0.5,\u00a0<\/p>\n\n\n\n<p>then sin(60\u00b0) = 2 \u2715 0.5 \u2715 0.866 = 0.866.<\/p>\n\n\n\n<p class=\"has-very-light-gray-to-cyan-bluish-gray-gradient-background has-background\"><strong>Also Read: <\/strong><strong>Branches of Mathematics: Arithmetic, Algebra, Geometry, Calculus, Trigonometry, Topology, Probability and Statistics<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-sum-to-product-identities-nbsp\">Sum to Product Identities\u00a0<\/h2>\n\n\n\n<p>The Sum to Product Identities allows the conversion of sums and differences of Trigonometric functions into products:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>sin(A) + sin(B) = 2sin((A+B)\/2)cos((A-B)\/2)<\/li>\n<\/ol>\n\n\n\n<ol class=\"wp-block-list\" start=\"2\">\n<li>sin(A) \u2013 sin(B) = 2cos((A+B)\/2)sin((A-B)\/2)<\/li>\n<\/ol>\n\n\n\n<ol class=\"wp-block-list\" start=\"3\">\n<li>cos(A) + cos(B) = 2cos((A+B)\/2)cos((A-B)\/2)<\/li>\n<\/ol>\n\n\n\n<ol class=\"wp-block-list\" start=\"4\">\n<li>cos(A) \u2013 cos(B) = -2sin((A+B)\/2)sin((A-B)\/2)<\/li>\n<\/ol>\n\n\n\n<p><strong>For Example:<\/strong>\u00a0<\/p>\n\n\n\n<p>If sin(30\u00b0) = 0.5 and sin(45\u00b0) = 0.707,\u00a0<\/p>\n\n\n\n<p>then sin(30\u00b0) + sin(45\u00b0) = 2 \u2715 sin(75\u00b0\/2) \u2715 cos(15\u00b0\/2) = 0.866 \u2715 0.966 = 0.837.<\/p>\n\n\n\n<p class=\"has-very-light-gray-to-cyan-bluish-gray-gradient-background has-background\"><strong>Also Read: <\/strong><strong>Profit and Loss Formula Questions<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-inverse-trigonometry-formulas-nbsp\">Inverse Trigonometry Formulas\u00a0<\/h2>\n\n\n\n<p>The inverse trigonometric functions, denoted as sin^-1, cos^-1, and tan^-1, allow us to find the angle given the value of the trigonometric function:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>sin^-1(x) = the angle whose sine is x<\/li>\n\n\n\n<li>cos^-1(x) = the angle whose cosine is x<\/li>\n\n\n\n<li>tan^-1(x) = the angle whose tangent is x<\/li>\n<\/ol>\n\n\n\n<p><strong>For Example:<\/strong>\u00a0<\/p>\n\n\n\n<p>If sin(x) = 0.5,\u00a0<\/p>\n\n\n\n<p>then x = sin^-1(0.5) = 30\u00b0.<\/p>\n\n\n\n<p class=\"has-very-light-gray-to-cyan-bluish-gray-gradient-background has-background\"><strong>Also Read: <\/strong><strong>20 Most Famous Indian Mathematicians<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-sine-law-nbsp-nbsp\">Sine Law\u00a0\u00a0<\/h2>\n\n\n\n<p>The Sine Law states that the ratio of the length of a side to the sine of the opposite angle is constant for all triangles:<\/p>\n\n\n\n<p>sin(A)\/a = sin(B)\/b = sin(C)\/c<\/p>\n\n\n\n<p><strong>For Example:<\/strong>\u00a0<\/p>\n\n\n\n<p>In a triangle with sides a = 5, b = 6, and c = 7, and angles A, B, and C, we can use the sine law to find the unknown angle A:<\/p>\n\n\n\n<p>sin(A)\/5 = sin(B)\/6<\/p>\n\n\n\n<p>sin(A) = (5\/6)sin(B)<\/p>\n\n\n\n<p class=\"has-very-light-gray-to-cyan-bluish-gray-gradient-background has-background\"><strong>Also Read: <\/strong><strong>Algebra Questions<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-cosine-law\">Cosine Law<\/h2>\n\n\n\n<p>The Cosine Law relates the lengths of the sides of a triangle to the cosine of one of the angles:<\/p>\n\n\n\n<p>c^2 = a^2 + b^2 \u2013 2ab cos(C)<\/p>\n\n\n\n<p><strong>For Example:<\/strong>\u00a0<\/p>\n\n\n\n<p>In a triangle with sides a = 5, b = 6, and c = 7, we can use the cosine law to find the angle C:<\/p>\n\n\n\n<p>7^2 = 5^2 + 6^2 \u2013 2(5)(6)cos(C)<\/p>\n\n\n\n<p>cos(C) = (5^2 + 6^2 \u2013 7^2) \/ (2 * 5 * 6) = 0.5<\/p>\n\n\n\n<p class=\"has-text-align-center has-vivid-red-color has-text-color has-link-color has-medium-font-size wp-elements-b9eb360a4bc4bdaa3c4feb84a1bd0d67\"><strong>Related Blogs\u00a0<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><tbody><tr><td><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/basic-concepts-types-of-fractions\/\"><strong>7 Types of Fractions with Examples<\/strong><\/a><\/td><td><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/basic-concepts-what-are-co-prime-numbers\/\"><strong>What are Co Prime Numbers?<\/strong><\/a><\/td><\/tr><tr><td><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/basic-concepts-how-to-find-percentage-of-marks\/\"><strong>How to Find Percentage of Marks?<\/strong><\/a><\/td><td><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/basic-concepts-tables-1-to-20\/\"><strong>Multiplication Tables of 1 to 20<\/strong><\/a><\/td><\/tr><tr><td><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/basic-concepts-ordinal-numbers\/\"><strong>Ordinal Numbers from 1 to 100!<\/strong><\/a><strong>\u00a0<\/strong><\/td><td><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/civics-and-polity-table-of-17\/\"><strong>Table of 17: Multiples up to 20 & a Trick!<\/strong><\/a>\u00a0\u00a0<\/td><\/tr><tr><td><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/basic-concepts-table-of-12\/\"><strong>Table of 12: Multiples up to 20!<\/strong><\/a><\/td><td><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/basic-concept-hcf-of-two-consecutive-odd-numbers\/\"><strong>What is the HCF of Two Consecutive Odd Numbers?<\/strong><\/a><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>I hope this helps! Did you like learning about Trigonometry Formulas? Keep reading our blogs to learn more about the <a href=\"https:\/\/leverageedu.com\/discover\/category\/school-education\/basic-concepts\/\"><strong>Basic Concepts of Maths<\/strong><\/a>!<\/p>\n","protected":false},"excerpt":{"rendered":"Trigonometry is the study of relationships between angles and sides of triangles and is a fundamental branch of&hellip;\n","protected":false},"author":106,"featured_media":825741,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"editor_notices":[],"footnotes":""},"categories":[423,476,389],"tags":[],"class_list":{"0":"post-825729","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-basic-concepts","8":"category-maths","9":"category-school-education"},"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v27.5 (Yoast SEO v27.5) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>Trigonometry Formulas, Examples and Solutions!\u00a0 - 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