If $x$ satisfies the equation $$\tan x = \dfrac{\sin 10^\circ + \sin 40^\circ}{ \cos 10^\circ + \cos 40^\circ}$$ and $x$ is between $0^\circ$ and $90^\circ$, then $x$ is equal to what?
Is there an identity I can use here?
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Blue
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2What did you try? Where did you get stuck? – Noah Schweber Nov 13 '16 at 03:32
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Yes, there is a directly relevant identity. It’s a half-angle formula for one of the trig functions. – Brian M. Scott Nov 13 '16 at 03:38
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http://mathworld.wolfram.com/ProsthaphaeresisFormulas.html – lab bhattacharjee Nov 13 '16 at 03:55
2 Answers
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Try this in the numerator:
$$\sin a + \sin b = 2 \sin \frac{a+b}2 \;\cos \frac{a-b}2$$
And this in the denominator:
$$\cos a + \cos b = 2 \cos \frac{a+b}2\;\cos\frac{a-b}2$$
Divide the numerator by the denominator after applying these sum-product relations, cancel out common factors, and see what you get.
Blue
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Oscar Lanzi
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(+1) The identity that ensues is one of my favorites. It also shows that $\tan\left(\frac a2\right)=\frac{\sin(a)}{1+\cos(a)}$ – robjohn Nov 13 '16 at 04:56
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Proof Without Words
$$ \large\color{#8060A0}{\frac{\sin(a)+\sin(b)}{\cos(a)+\cos(b)}=\tan\left(\frac{a+b}2\right)} $$
robjohn
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