{"id":9452,"date":"2022-01-14T16:22:57","date_gmt":"2022-01-14T10:52:57","guid":{"rendered":"https:\/\/mathemerize.com\/?p=9452"},"modified":"2022-01-14T16:22:59","modified_gmt":"2022-01-14T10:52:59","slug":"what-is-the-general-solution-of-cos-theta-cos-alpha","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/what-is-the-general-solution-of-cos-theta-cos-alpha\/","title":{"rendered":"What is the General Solution of \\(cos \\theta\\) = \\(cos \\alpha\\) ?"},"content":{"rendered":"
Solution :<\/h2>\n
The general solution of \\(cos \\theta\\) = \\(cos \\alpha\\)<\/strong> is given by \\(\\theta\\) = \\(2n\\pi \\pm \\alpha\\), n \\(\\in\\) Z.<\/strong><\/p>\n
Proof :<\/strong><\/p>\n
We have, \\(cos \\theta\\) = \\(cos \\alpha\\)<\/p>\n
\\(\\implies\\) \\({\\theta + \\alpha\\over 2}\\) = \\(n\\pi\\) or \\({\\theta – \\alpha\\over 2}\\) = \\(n\\pi\\) , n \\(\\in\\) Z<\/p>\n
\\(\\implies\\) \\(\\theta\\) = \\(2n\\pi – \\alpha\\), \\(\\in\\) Z or, \\(\\theta\\) = \\(2n\\pi + \\alpha\\), n \\(\\in\\) Z.<\/p>\n
\\(\\implies\\) \\(\\theta\\) = \\(2n\\pi \\pm \\alpha\\), where n \\(\\in\\) Z.<\/p>\n
Remark : <\/strong>The equation \\(sec \\theta\\) = \\(sec \\alpha\\) is equivalent to \\(cos \\theta\\) = \\(cos \\alpha\\). Thus, \\(sec \\theta\\) = \\(sec \\alpha\\) and \\(cos \\theta\\) = \\(cos \\alpha\\) have the same general solution.<\/p>\n","protected":false},"excerpt":{"rendered":"
Solution : The general solution of \\(cos \\theta\\) = \\(cos \\alpha\\) is given by \\(\\theta\\) = \\(2n\\pi \\pm \\alpha\\), n \\(\\in\\) Z. Proof : We have, \\(cos \\theta\\) = \\(cos \\alpha\\) \\(\\implies\\) \\(cos \\theta\\) – \\(cos \\alpha\\) = 0 \\(\\implies\\) -\\(2 sin ({\\theta + \\alpha\\over 2}) sin({\\theta – \\alpha\\over 2})\\) = 0 \\(\\implies\\) \\(sin ({\\theta …<\/p>\n