{"id":154796,"date":"2022-03-25T23:56:00","date_gmt":"2022-03-25T18:26:00","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/magic-hexagon-for-trig-identities-building-derivation-and-problems\/"},"modified":"2023-01-10T18:03:01","modified_gmt":"2023-01-10T12:33:01","slug":"magic-hexagon-for-trig-identities-building-derivation-and-problems","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/magic-hexagon-for-trig-identities\/","title":{"rendered":"Magic Hexagon for Trig Identities &#8211; Building, Derivation and Problems"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/magic-hexagon-for-trig-identities\/#Trig_Magic_Hexagon\" title=\"Trig Magic Hexagon\">Trig Magic Hexagon<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/magic-hexagon-for-trig-identities\/#Normal_Magical_Hexagons\" title=\"Normal Magical Hexagons\">Normal Magical Hexagons<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/magic-hexagon-for-trig-identities\/#Trigonometry_Hexagon\" title=\"Trigonometry Hexagon\">Trigonometry Hexagon<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/magic-hexagon-for-trig-identities\/#Building_the_Trig_Hexagon_Identities\" title=\"Building the Trig Hexagon Identities\">Building the Trig Hexagon Identities<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/magic-hexagon-for-trig-identities\/#Pythagorean_Identities\" title=\"Pythagorean Identities\">Pythagorean Identities<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/magic-hexagon-for-trig-identities\/#Deriving_the_Pythagorean_Identities\" title=\"Deriving the Pythagorean Identities\">Deriving the Pythagorean Identities<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/magic-hexagon-for-trig-identities\/#Even_and_Odd_Identities\" title=\"Even and Odd Identities\">Even and Odd Identities<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Trig_Magic_Hexagon\"><\/span>Trig Magic Hexagon<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>A trig magic hexagon is a hexagon-shaped array of trigonometric functions that are related to each other. The six trigonometric functions that are typically used in a magic hexagon are sine, cosine, tangent, cosecant, secant, and cotangent. Each of the six functions is related to two of the others, and these relationships are shown in the hexagon. The relationships are as follows:<\/p>\n<p>sine = cosine<br \/>\ntangent = secant<br \/>\ncosecant = cotangent<br \/>\ncosecant = secant<br \/>\ncosine = tangent<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-154795 size-medium\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/magic-hexagon-for-trig-identities-building-derivation-and-problems-300x212.jpg\" alt=\"Magic Hexagon for Trig Identities\" width=\"300\" height=\"212\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/magic-hexagon-for-trig-identities-building-derivation-and-problems-300x212.jpg?v=1648232754 300w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/magic-hexagon-for-trig-identities-building-derivation-and-problems.jpg?v=1648232754 606w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Normal_Magical_Hexagons\"><\/span>Normal Magical Hexagons<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Normal magical hexagons are hexagons that are not associated with any specific magical properties. They are typically used for decorative purposes or in spells that do not require any specific magical properties.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Trigonometry_Hexagon\"><\/span>Trigonometry Hexagon<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>A hexagon is a six-sided polygon. It has six angles and six sides. The angles of a hexagon can be found by using the trigonometric functions sine, cosine, and tangent. The length of the sides of a hexagon can also be found using the trigonometric functions sine, cosine, and tangent.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Building_the_Trig_Hexagon_Identities\"><\/span>Building the Trig Hexagon Identities<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The trigonometric functions are periodic, which means that they repeat over and over again. The period of a function is the amount of time it takes for the function to repeat.<\/p>\n<p>The trigonometric functions have a period of 360 degrees. This means that they repeat every 360 degrees.<\/p>\n<p>The trigonometric functions can be used to find the exact value of a trigonometric function for any angle. However, not all angles can be written in standard form.<\/p>\n<p>Standard form is when an angle is written in terms of its radian measure. Radian measure is the number of degrees in a circle divided by the number of radians in a circle. There are 2\u03c0 radians in a circle.<\/p>\n<p>There are a few angles that can be written in standard form, and these are the angles that we will be working with in this section.<\/p>\n<p>The six angles that can be written in standard form are: 0\u00b0, 30\u00b0, 45\u00b0, 60\u00b0, 90\u00b0, and 180\u00b0.<\/p>\n<p>The Trig Hexagon<\/p>\n<p>The trigonometric functions can be used to find the exact value of a trigonometric function for any angle. However, not all angles can be written in standard form.<\/p>\n<p>Standard form is when an angle is written in terms of its radian measure. Radian measure is the number of degrees in a circle divided by the number of radians in a circle. There are 2\u03c0 rad<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Pythagorean_Identities\"><\/span>Pythagorean Identities<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>There are a few Pythagorean Identities that are worth memorizing.<\/p>\n<p>1. The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.<\/p>\n<p>2. The cosine of an angle is equal to the ratio of the length of the adjacent side to the length of the hypotenuse.<\/p>\n<p>3. The sine of an angle is equal to the ratio of the length of the opposite side to the length of the hypotenuse.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Deriving_the_Pythagorean_Identities\"><\/span>Deriving the Pythagorean Identities<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The Pythagorean Identities are a set of three mathematical identities that are named after the ancient Greek mathematician Pythagoras. These identities relate the Pythagorean theorem to other mathematical concepts.<\/p>\n<p>The Pythagorean theorem states that in a right angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. The Pythagorean Identities allow us to relate this theorem to other mathematical concepts.<\/p>\n<p>The first Pythagorean identity states that the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides, plus the square of the length of the hypotenuse.<\/p>\n<p>The second Pythagorean identity states that the square of the length of the hypotenuse is equal to the product of the squares of the other two sides, minus the square of the length of the hypotenuse.<\/p>\n<p>The third Pythagorean identity states that the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides, minus the square of the length of the longest side.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Even_and_Odd_Identities\"><\/span>Even and Odd Identities<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>An odd identity is an equation that states that the sum of two odd numbers is an odd number. An even identity is an equation that states that the sum of two even numbers is an even number.<\/p>\n<p>The sum of two odd numbers is always an odd number.<\/p>\n<p>The sum of two even numbers is always an even number.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Trig Magic Hexagon A trig magic hexagon is a hexagon-shaped array of trigonometric functions that are related to each other. 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