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2011-02-01T00:00:01
<mdoxml version="1.0"><div>Draw a right-angled triangle with angles $90^\circ$,
$(45+x)^\circ$ and $(45-x)^\circ$.</div>
<br></br>
<div>What does Pythagoras' Theorem tell you about these
angles?</div>
<br></br>
<div>Use this information to find $ \sin^2 1^\circ + \sin^2 2^\circ
+ \, \cdots \,+ \sin^2 359 ^\circ + \sin^2 360^\circ$.</div>
<br></br></mdoxml>
<mdoxml version="1.0"><br></br>
<mdo:image alt="" src="degree1.gif"></mdo:image> <br></br>
<br></br>
In this diagram the right angled triangle has hypotenuse of length
1 unit so the lengths of the sides are $\sin(45-x)^o$ and
$\sin(45+x)^o$. By Pythagoras Theorem: $$ \sin^2(45-x)^o +
\sin^2(45+x)^o = 1.$$ These pairings of values that add up to 1 are
useful in evaluating the expression: $$ A = \sin^2 1^o + \sin^2 2^o
+ ... + \sin^2 359^o + \sin^2 360^\circ.$$<br></br>
<br></br>
Ella Ryan from Madras College, St Andrew's based her solution on
the symmetries of the graph of $y=\sin^2 x$. <br></br>
<br></br>
<mdo:image alt="" src="degree2.gif"></mdo:image><br></br>
Consider the graph of $y=\sin^2 x^o$ between $x=1$ and $x=89$
inclusive and pairs of points having $y$ values which, when added
together, always equal one. This result is equivalent to Pythagoras
Theorem as explained above. For example, $$\sin^2 50^o + \sin^2
40^o = 1.$$ The pairs of points can be labelled $(45-x)^o$ and
$(45+x)^o$ or alternatively $x^o$ and $(90-x)^o$.<br></br>
<br></br>
Essentially the same method was used both by Ella and also by Hou
Yang Yang, Millfield School, Somerset, U.K.<br></br>
<br></br>
Firstly we use the following symmetry properties of the sine
function: $$ \sin^2 1^o = \sin^2 179^0 = \sin^2 181^o = \sin^2
359^o,$$ $$ \sin^2 2^o = \sin^2 178^o = \sin^2 182^o = \sin^2
358^o, ...$$ $$ \sin^2 89^o = \sin^2 91^o = \sin^2 269^o =\sin^2
271^o$$ and also $\sin^2 90^o = \sin^2 270^o = 1 $ and $\sin 180^o
= \sin 360^o = 0.$ Pairing equal values in $A$ gives:<br></br>
<br></br>
<div class="math">\begin{eqnarray} A &=& 2(\sin^2 1^o +
\sin^2 2^o + ... + \sin^2 179^o),\\ &=& 4(\sin^2 1^o +
\sin^2 2^o + ... + \sin^2 89^o) + 2\sin^2 90^o .
\end{eqnarray}</div>
<br></br>
<br></br>
Then taking pairs that add up to 1 we get:
<div class="math">\begin{eqnarray} A &=& 4[(\sin^2 1^o +
\sin^2 89^o) + (\sin^2 2^o + \sin^2 88^o) + ... \\ && +
(\sin^2 44^o + \sin^2 46^o) + \sin^2 45^o] + 2\\ &=& 4[44 +
0.5] + 2 \\ &=& 180. \end{eqnarray}</div>
<br></br></mdoxml>
<mdoxml version="1.0"><br></br>
<p>You can draw the graph of the sine function. What about the graph of $y = \sin^2 x$? What about the periodicity of this graph?</p>
<p> </p>
<p>What can you say about the angles (45 + x) degrees and (45 - x) degrees, the sines of these angles and the squares of the sines of these angles?</p></mdoxml>
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Degree Ceremony
What does Pythagoras' Theorem tell you about these angles:
90°, (45+x)° and (45-x)° in a triangle? Find sin^2
1° + sin^2 2° + ... + sin^2 359 ° + sin^2 360°.
Trigonometry
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Trigonometry
Sine
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