{"id":2881,"date":"2022-09-26T13:46:00","date_gmt":"2022-09-26T17:46:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=2881"},"modified":"2024-10-25T11:00:12","modified_gmt":"2024-10-25T15:00:12","slug":"trigonometry-challenge-2022-09-26","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=2881","title":{"rendered":"Trigonometry Challenge 2022\/09\/26"},"content":{"rendered":"\n

Let $\\alpha$, $\\beta$, and $\\gamma$ be the interior angles of $\\triangle{ABC}$. Find all solutions so that $$\\cos\\alpha\\cdot\\cos\\beta+\\sin\\alpha\\cdot\\sin\\beta\\cdot\\sin\\gamma=1$$<\/p>\n\n\n\n

\ud83d\udd11<\/a><\/p>\n

\n\n\n\n

Solution<\/strong>: Since $\\alpha$, $\\beta$, and $\\gamma$ are the interior angles of $\\triangle{ABC}$, we have $$0\\lt\\alpha,\\ \\beta,\\ \\gamma\\lt\\pi$$ Therefore $$0\\lt\\sin\\alpha,\\ \\sin\\beta,\\ \\sin\\gamma\\le 1$$ and $$\\sin\\alpha\\cdot\\sin\\beta\\cdot\\sin\\gamma\\lt1$$<\/p>\n\n\n\n

If $$\\cos\\alpha\\cdot\\cos\\beta+\\sin\\alpha\\cdot\\sin\\beta\\cdot\\sin\\gamma=1 \\tag{1}$$ then $$\\cos\\alpha\\cdot\\cos\\beta\\gt0$$ Therefore $$0\\lt\\alpha,\\ \\beta\\le\\dfrac{\\pi}{2}$$ as $\\cos\\alpha$ and $\\cos\\beta$ cannot be both negative at the same time, which would result in $\\dfrac{\\pi}{2}\\lt\\alpha,\\ \\beta\\lt\\pi$ and $\\gamma\\lt 0$.<\/p>\n\n\n\n

Since all terms in equation (1) are positive, and $\\sin\\gamma\\le 1$, we have $$\\cos\\alpha\\cdot\\cos\\beta+\\sin\\alpha\\cdot\\sin\\beta\\cdot\\sin\\gamma\\le \\cos\\alpha\\cdot\\cos\\beta+\\sin\\alpha\\cdot\\sin\\beta\\cdot=\\cos(\\alpha-\\beta)$$<\/p>\n\n\n\n

Since $\\cos(\\alpha-\\beta)\\le1$, we have $$\\cos\\alpha\\cdot\\cos\\beta+\\sin\\alpha\\cdot\\sin\\beta\\cdot\\sin\\gamma\\le1$$<\/p>\n\n\n\n

The equal sign applies only if $\\cos(\\alpha-\\beta)=1$ and $\\sin\\gamma=1$. This implies that the solution for equation (1) is $$\\boxed{\\alpha=\\beta=\\dfrac{\\pi}{4},\\ \\gamma=\\dfrac{\\pi}{2}}$$ i.e. $\\triangle{ABC}$ is an isosceles right triangle.<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Let $\\alpha$, $\\beta$, and $\\gamma$ be the interior angles of $\\triangle{ABC}$. Find all solutions so that $$\\cos\\alpha\\cdot\\cos\\beta+\\sin\\alpha\\cdot\\sin\\beta\\cdot\\sin\\gamma=1$$ \ud83d\udd11 Solution: Since $\\alpha$, $\\beta$, and $\\gamma$ are the interior angles of $\\triangle{ABC}$, we have $$0\\lt\\alpha,\\ \\beta,\\ \\gamma\\lt\\pi$$ Therefore $$0\\lt\\sin\\alpha,\\ \\sin\\beta,\\ \\sin\\gamma\\le 1$$ … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[14,15],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2881"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2881"}],"version-history":[{"count":16,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2881\/revisions"}],"predecessor-version":[{"id":4599,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2881\/revisions\/4599"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2881"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2881"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2881"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}