Question

@callisto, help here please:)

Help here please:

Find the exact value of the following, show all work:

a. sin^2(75°) / sin 45° cos^2 30° (Hint: sin^2 θ = (sin θ)^2):

Answer 1

Not an easy one ...
cos2θ = 2cos^θ-1
Now put θ=15°
cos 2(15°) = 2cos^2 15° -1
cos 30 = 2cos^ 15° -1
sqrt(3) /2 = 2cos^2 15° -1
(sqrt3 + 2) /2 = 2cos^215°
cos15° = sqrt{(sqrt3 + 2)} /2
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\[sin^2(75°) / ( sin 45° cos^2 30° )\]\[= cos ^2(90°-75°)/ ( sin 45° cos^2 30° )\]\[= cos ^2(15°)/ ( sin 45° cos^2 30° )\]\[= [\sqrt{(\sqrt3 + 2)} /2]^2/ [(\sqrt2/2) (\sqrt3 /2 )^2]\]\[= [(\sqrt3 + 2 /4)/ [(\sqrt2/2) (\sqrt3 /2 )^2]\]\[= [(\sqrt3 + 2 /4)/ [(\sqrt2/2) (3 /4 )]\]\[= [(\sqrt3 + 2 /4)/ [(3\sqrt2/8)]\]\[= 2(\sqrt3 + 2 )/ (3\sqrt2)\]\[= 2\sqrt2(\sqrt3 + 2 )/ (3\times2)\]\[= \sqrt2(\sqrt3 + 2 )/ (3)\]\[= (\sqrt6 + 2\sqrt2 )/ (3)\]

Sorry , just woke up :(

Answer 2

Omg, Thank you so much, that must have been miserable to type. Can you give me a couple of minutes to analyse this? Can I ask you questions after a couple of mins?

Answer 3

Welcome...my method is a little....stupid...perhaps..... SORRY!!!!!

Answer 4

I went this route but stopped before I did the denominator part.
I was trying to avoid that half-angle formula.

sin(A+B)=sin A cos B + cos A sin B

sin(75) = sin(30 + 45) = sin 30 *cos 45 + cos 30* sin 45
sin(75) = (1/2) * sqrt(2)/2 + sqrt(3)/2*sqrt(2)/2
sin (75) = [ sqrt(2) + sqrt(6) /4 ]
[sin (75) ] ^2 = [ sqrt(2) + sqrt(6) /4 ]*[ sqrt(2) + sqrt(6) /4 ]

Answer 5

Yeah, I got that numerator already:)

Answer 6

weird part is solving the denominator and then simplifying

Answer 7

@Open2study -> the denominator is 3 sqrt(2)/8, isn't it?

Answer 8

Yes it is

Answer 9

I can't read what Callisto got because the translator or whatever is not working.

Answer 10

Sorry, that's why I said my method was... stupid :(

Answer 11

@callisto, Dont say that! your methods always help me get great answers:)

Answer 12

Not this time though

Answer 13

@Callisto --> It's not stupid. It's this program that converts MathLab to text or AjaxMath or something.

Answer 14

when answering this type of questions, do I continue untill i get simplest form?

Answer 15

Numerator: [ sqrt(2) + sqrt(6) /4 ]*[ sqrt(2) + sqrt(6) /4 ]

Denominator: 3 sqrt(2)/8

Trig is always tedious like this.

Answer 16

So what can we simplify?

Answer 17

the denominators, 4, 4, and 8

Answer 18

right?

Answer 19

The numerator when squared out is: 19/8 + sqrt(3) or [19 + 8 sqrt(3)] / 8