{"appState":{"pageLoadApiCallsStatus":true},"articleState":{"article":{"headers":{"creationTime":"2016-03-26T20:24:12+00:00","modifiedTime":"2022-08-10T20:48:46+00:00","timestamp":"2022-09-14T18:19:53+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Trigonometry","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33729"},"slug":"trigonometry","categoryId":33729}],"title":"Express Sine in Terms of Cotangent","strippedTitle":"express sine in terms of cotangent","slug":"express-sine-in-terms-of-cotangent","canonicalUrl":"","seo":{"metaDescription":"This trigonometry article provides step-by-step instructions for how to express sine in terms of cotangent.","noIndex":0,"noFollow":0},"content":"Even though each trigonometry function is perfectly wonderful, being able to express each trig function in terms of one of the other five trig functions is frequently to your advantage. For example, you may have some sine terms in an expression that you want to express in terms of cotangent, so that all the functions match, making it easier to solve the equation.\r\n\r\nTo write the sine function in terms of cotangent, follow these steps:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Start with the ratio identity involving sine, cosine, and tangent, and multiply each side by cosine to get the sine alone on the left.</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/274246.image0.png\" alt=\"The ratio identity tangent equals sine over cosine.\" width=\"117\" height=\"67\" /></li>\r\n \t<li>\r\n<p class=\"first-para\">Replace cosine with its reciprocal function.</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/274247.image1.png\" alt=\"Sine in terms of tangent and secant.\" width=\"129\" height=\"48\" /></li>\r\n \t<li>\r\n<p class=\"first-para\">Solve the Pythagorean identity tan<sup>2</sup>θ + 1 = sec<sup>2</sup>θ for secant.</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/274248.image2.png\" alt=\"Solving the Pythagorean identity squared tangent plus one equals squared secant.\" width=\"293\" height=\"31\" /></li>\r\n \t<li>\r\n<p class=\"first-para\">Replace the secant in the sine equation.</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/274249.image3.png\" alt=\"Replacing the secant in the sine equation.\" width=\"383\" height=\"56\" /></li>\r\n \t<li>\r\n<p class=\"first-para\">Replace all the tangents with 1 over the reciprocal for tangent (which is cotangent) and simplify the expression.</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/274250.image4.png\" alt=\"Expressing the sine in terms of cotangent.\" width=\"131\" height=\"85\" />\r\n<p class=\"child-para\">The result is a <i>complex fraction</i> — it has fractions in both the numerator and denominator — so it’ll look a lot better if you simplify it.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Rewrite the part under the radical as a single fraction and simplify it by taking the square root of each part.</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/274251.image5.png\" alt=\"Simplify the expression of the sine in terms of cotangent.\" width=\"348\" height=\"189\" /></li>\r\n \t<li>\r\n<p class=\"first-para\">Multiply the numerator by the reciprocal of the denominator.</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/274252.image6.png\" alt=\"Formula for sine in terms of cotangent.\" width=\"339\" height=\"85\" />\r\n<p class=\"child-para\">Voilà — you have sine in terms of cotangent.</p>\r\n</li>\r\n</ol>","description":"Even though each trigonometry function is perfectly wonderful, being able to express each trig function in terms of one of the other five trig functions is frequently to your advantage. For example, you may have some sine terms in an expression that you want to express in terms of cotangent, so that all the functions match, making it easier to solve the equation.\r\n\r\nTo write the sine function in terms of cotangent, follow these steps:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Start with the ratio identity involving sine, cosine, and tangent, and multiply each side by cosine to get the sine alone on the left.</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/274246.image0.png\" alt=\"The ratio identity tangent equals sine over cosine.\" width=\"117\" height=\"67\" /></li>\r\n \t<li>\r\n<p class=\"first-para\">Replace cosine with its reciprocal function.</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/274247.image1.png\" alt=\"Sine in terms of tangent and secant.\" width=\"129\" height=\"48\" /></li>\r\n \t<li>\r\n<p class=\"first-para\">Solve the Pythagorean identity tan<sup>2</sup>θ + 1 = sec<sup>2</sup>θ for secant.</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/274248.image2.png\" alt=\"Solving the Pythagorean identity squared tangent plus one equals squared secant.\" width=\"293\" height=\"31\" /></li>\r\n \t<li>\r\n<p class=\"first-para\">Replace the secant in the sine equation.</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/274249.image3.png\" alt=\"Replacing the secant in the sine equation.\" width=\"383\" height=\"56\" /></li>\r\n \t<li>\r\n<p class=\"first-para\">Replace all the tangents with 1 over the reciprocal for tangent (which is cotangent) and simplify the expression.</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/274250.image4.png\" alt=\"Expressing the sine in terms of cotangent.\" width=\"131\" height=\"85\" />\r\n<p class=\"child-para\">The result is a <i>complex fraction</i> — it has fractions in both the numerator and denominator — so it’ll look a lot better if you simplify it.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Rewrite the part under the radical as a single fraction and simplify it by taking the square root of each part.</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/274251.image5.png\" alt=\"Simplify the expression of the sine in terms of cotangent.\" width=\"348\" height=\"189\" /></li>\r\n \t<li>\r\n<p class=\"first-para\">Multiply the numerator by the reciprocal of the denominator.</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/274252.image6.png\" alt=\"Formula for sine in terms of cotangent.\" width=\"339\" height=\"85\" />\r\n<p class=\"child-para\">Voilà — you have sine in terms of cotangent.</p>\r\n</li>\r\n</ol>","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. 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Express Sine in Terms of Cotangent

By: Mary Jane Sterling and

Updated: 08-10-2022

Trigonometry For Dummies

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Even though each trigonometry function is perfectly wonderful, being able to express each trig function in terms of one of the other five trig functions is frequently to your advantage. For example, you may have some sine terms in an expression that you want to express in terms of cotangent, so that all the functions match, making it easier to solve the equation.

To write the sine function in terms of cotangent, follow these steps:

Start with the ratio identity involving sine, cosine, and tangent, and multiply each side by cosine to get the sine alone on the left.
Replace cosine with its reciprocal function.
Solve the Pythagorean identity tan²θ + 1 = sec²θ for secant.
Replace the secant in the sine equation.
Replace all the tangents with 1 over the reciprocal for tangent (which is cotangent) and simplify the expression.

The result is a complex fraction — it has fractions in both the numerator and denominator — so it’ll look a lot better if you simplify it.
Rewrite the part under the radical as a single fraction and simplify it by taking the square root of each part.
Multiply the numerator by the reciprocal of the denominator.

Voilà — you have sine in terms of cotangent.

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