240 Followers Problem - 2

Geometry Level 4

cos ⁡ 2 θ 1 + cos ⁡ 2 θ 2 + cos ⁡ 2 θ 3 \large \cos^2 \theta_{1} + \cos^2 \theta_{2} +\cos^2 \theta_{3}

Given that cos ⁡ ( θ 1 − θ 2 ) + cos ⁡ ( θ 2 − θ 3 ) + cos ⁡ ( θ 3 − θ 1 ) = − 3 2 \cos \left( \theta_{1} -\theta_{2} \right) + \cos \left( \theta_{2} -\theta_{3} \right) + \cos \left( \theta_{3} -\theta_{1} \right) = -\dfrac{3}{2} , find the value of the expression above.


The answer is 1.5000000.

View wiki
Surya Prakash by Surya Prakash , 22, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

T h e s u m C o s ( θ 1 − θ 2 ) + C o s ( θ 2 − θ 3 ) + C o s ( θ 3 − θ 1 ) i s − t i v e = − 1 1 2 . Three terms can be split in the following groups:- ( − 1 / 2 , − 1 / 2 , − 1 / 2 ) , ( − 1 / 2 , 0 , − 1 ) , ( − 1 , − 1 , 1 / 2 ) . The following only can give rational Cosine and Sine for rational angles. Angles with |CosX|= 1/2, are (60,120,240,300), … … |CosX|=0, are (90,270), and |CosX|=1, are (0,180). Examining the first group, (-1/2, -1/2, -1/2):- To get - 1/2 for the Cos(difference), the difference must be 120 or 240. ⟹ Two left angles may be two of the angles. Let us check. θ 1 = 60 , θ 2 = 300. ⟹ θ 1 − θ 2 = − 240 = 120. O K . θ 2 − θ 3 = 120 o r 240. ⟹ θ 3 = 300 − 120 = 180 O K o r 300 − 240 = 60 N O T g o o d . ⟹ θ 1 = 6 0 o , θ 2 = 30 0 o , θ 3 = 180. ∴ C o s 2 θ 1 + C o s 2 θ 2 + C o s 2 θ 3 = 1 4 + 1 4 + 1 = 1.5 The~ sum~~ Cos(\theta_1-\theta_2)+ Cos(\theta_2-\theta_3)+ Cos(\theta_3-\theta_1) ~ is ~ -tive ~ = - 1\frac 1 2.\\ \text{Three terms can be split in the following groups:-}\\ (-1/2, -1/2, -1/2) ,~~~~~ (-1/2, 0, -1) ,~~~~~ (-1, -1, 1/2) .\\ \text{The following only can give rational Cosine and Sine for rational angles.}\\ \text{Angles with |CosX|= 1/2, are (60,120,240,300),……|CosX|=0, are (90,270),}\\ \text{and |CosX|=1, are (0,180).}\\ \text {Examining the first group, (-1/2, -1/2, -1/2):-}\\ \text{To get - 1/2 for the Cos(difference), the difference must be 120 or 240.}\\ \implies~ \text{Two left angles may be two of the angles. Let us check.}\\ \theta_1=60,~~\theta_2=300. \implies~\theta_1 - \theta_2= - 240=120. ~~~~OK.\\ \theta_2 - \theta_3=120~~or~~240. \implies~\theta_3=300-120=180~~OK~~or~~300-240=60~~NOT~good.\\ \implies~~~ \theta_1=60^o,~~~~ \theta_2=300^o,~~~~ \theta_3=180.\\ \therefore~~~Cos^2\theta_1+ Cos^2\theta_2+ Cos^2\theta_3 =\dfrac 1 4 + \dfrac 1 4 + 1 =\Large ~~~~~~\color{#D61F06}{1.5}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Regarding the first line: why can't the differences be -1, -1 and 0.5?

In general your solution looks like a lot of guessing :)

Ton de Moree - 5 years, 7 months ago

Log in to reply

I am adding more explanations to my solution.

Niranjan Khanderia - 5 years, 7 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...