{"id":1109,"date":"2014-12-31T02:52:23","date_gmt":"2014-12-31T07:52:23","guid":{"rendered":"http:\/\/www.curtisbright.com\/bln\/?p=1109"},"modified":"2014-12-31T02:52:23","modified_gmt":"2014-12-31T07:52:23","slug":"a-difference-of-squared-sines","status":"publish","type":"post","link":"http:\/\/localhost\/blog\/index.php\/2014\/12\/31\/a-difference-of-squared-sines\/","title":{"rendered":"A difference of squared sines"},"content":{"rendered":"

The purpose of this post is to show\u00a0the interesting identity<\/p>\n

\\[ \\sin(x)^2-\\sin(y)^2 = \\sin(x+y)\\sin(x-y) . \\]<\/p>\n

<\/p>\n

In fact, this is a simple consequence of the sine addition identity<\/a><\/p>\n

\\[ \\sin(x\\pm y) = \\sin x\\cos y\\pm\\cos x\\sin y \\]<\/p>\n

and\u00a0the fundamental\u00a0Pythagorean identity<\/a> $\\sin(\\theta)^2+\\cos(\\theta)^2=1$.<\/p>\n

The demonstration is a pretty straightforward usage of the above identities, but involves a little bit of trickery—on the third step, we add and subtract the quantity $\\sin(x)^2\\sin(y)^2$:<\/p>\n

\\begin{align*}
\n\\sin(x+y)\\sin(x-y) &= (\\sin x\\cos y+\\cos x\\sin y)(\\sin x\\cos y-\\cos x\\sin y) \\\\
\n&= \\sin(x)^2\\cos(y)^2-\\cos(x)^2\\sin(y)^2 \\\\
\n&\\qquad+\\cos x\\sin y\\sin x\\cos y-\\cos x\\sin y\\sin x\\cos y \\\\
\n&= \\sin(x)^2\\cos(y)^2-\\cos(x)^2\\sin(y)^2 \\\\
\n&\\qquad+\\sin(x)^2\\sin(y)^2-\\sin(x)^2\\sin(y)^2 \\\\
\n&= \\sin(x)^2(\\cos(y)^2+\\sin(y)^2) \\\\
\n&\\qquad-\\sin(y)^2(\\cos(x)^2+\\sin(x)^2) \\\\
\n&= \\sin(x)^2-\\sin(y)^2
\n\\end{align*}<\/p>\n

This\u00a0was not an identity I knew off the top of my head, but it came up on a problem I was working on. We’ll have reason to use it in a later post, which is why I wanted to single it out right now.<\/p>\n","protected":false},"excerpt":{"rendered":"

The purpose of this post is to show\u00a0the interesting identity \\[ \\sin(x)^2-\\sin(y)^2 = \\sin(x+y)\\sin(x-y) . \\]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[],"_links":{"self":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/posts\/1109"}],"collection":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/comments?post=1109"}],"version-history":[{"count":0,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/posts\/1109\/revisions"}],"wp:attachment":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/media?parent=1109"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/categories?post=1109"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/tags?post=1109"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}