Find the value of:
(cos^4(75°) + sin^4(75°) + 3sin^2(75º)cos^2(75°)) / (cos^6(75º) + sin^6(75°) + 4sin^2(75°)cos^2 (75º))
I first found the values for sin75 and cos75. (skipping the degree symbol for easier typing)
sin75 = sin(30+45) = sin30cos45 + cos30sin45 = (1/2)(√2/2) + (√3/2)(√2/2) = ( √2 + √6 /4)
sin^2 75 = (√2+√6)^2/16 = (4 + 2√12 + 6)/16 = (8 + 4√3)/16 = (2 + √3)/4
sin^4 75 = [ (2+√3)/4 ]^2 = .... = (7 + 4√3)/16
sin^6 75 = (sin^2 75)(sin^4 75) = (2 + √3)/4 * (7 + 4√3)/16 = (26 + 15√3)/64
I then did the same for cos75 = cos30cos45 - sin30sin45 = (√3/2)(√2/2) - (1/2)(√2/2) = (√6 - √2)/4
cos^2 75 = [ (√6-√2)/4 ]^2 = (6 - 2√12 + 2)/16 = (8 - 4√3)/16 = (2 - √3)/4
cos^4 75 = [ (2 - √3)/4 )^2 = ....... = (7 - 4√3)/16
cos^6 75 = (cos^2 75)(cos^4 75) = [(2 + √3)/4][(2 - √3)/4] = (26-15√3)/64
and finally (sin^2 75)(cos^2 75) = [(2 + √3)/4][(2 - √3)/4) = 1/16
(notice the nice symmetry between the corresponding answers, that is useful,
almost done .....
(cos^4(75°) + sin^4(75°) + 3sin^2(75º)cos^2(75°)) / (cos^6(75º) + sin^6(75°) + 4sin^2(75°)cos^2 (75º))
= [ (7 - 4√3)/16 + (7 + 4√3)/16 + 3(1/16) ] / [(26-15√3)/64 + (26 + 15√3)/64 + 4(1/16 ]
= [ 17/16] / [68/64]
= (17/16)(64/68)
= 1
yeaahhhhh!!!! That was fun
I skipped several of the simplifications as you can see, however, they are all correct
I will leave it up to you to verify the missing ones.
(cos^4x + sin^4x + 3 sin^2x cos^2x)
--------------------------------------------------
(cos^6x + sin^6x + 4 sin^2x cos^2x)
= (-cos^4x + cos^2x + 1) / (-cos^4x + cos^2x + 1)
= 1
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To find the value of the given expression, let's break it down step by step:
First, let's recall the trigonometric identity for cos(2θ):
cos(2θ) = cos^2(θ) - sin^2(θ)
Using this identity, we can rewrite the expression as:
(cos^4(75°) + sin^4(75°) + 3sin^2(75º)cos^2(75°)) /
(cos^6(75º) + sin^6(75°) + 4sin^2(75°)cos^2 (75º))
Now, let's substitute sin^2(θ) = 1 - cos^2(θ) into the expression:
(cos^4(75°) + (1 - cos^2(75°))^2 + 3(1 - cos^2(75°))cos^2(75°)) /
(cos^6(75º) + (1 - cos^2(75°))^3 + 4(1 - cos^2(75°))cos^2 (75º))
Expanding the squares and cube in the numerator and denominator, we get:
(cos^4(75°) + (1 - 2cos^2(75°) + cos^4(75°)) + 3(1 - cos^2(75°))cos^2(75°)) /
(cos^6(75º) + (1 - 3cos^2(75°) + 3cos^4(75°) - cos^6(75°)) + 4(1 - cos^2(75°))cos^2 (75º))
Simplifying further:
(cos^4(75°) + 1 - 2cos^2(75°) + cos^4(75°) + 3cos^2(75°) - 3cos^4(75°)) /
(cos^6(75º) + 1 - 3cos^2(75°) + 3cos^4(75°) - cos^6(75°) + 4cos^2 (75º) - 4cos^4(75°))
Now, combining like terms:
(2cos^4(75°) + cos^2(75°) + 1) /
(3cos^4(75°) + 5cos^2(75°) + 1)
Finally, we can simplify further by dividing each term by cos^2(75°):
(2cos^2(75°) + 1 + cos^(-2)(75°)) /
(3cos^2(75°) + 5 + cos^(-2)(75°))
Now, we need to find the value of cos^2(75°). To do this, we can use the trigonometric identity for cos(2θ) we mentioned earlier:
cos(150°) = cos^2(75°) - sin^2(75°)
We know that cos(150°) = -√3/2 and sin(75°) = √(1 - cos^2(75°)). By substituting these values, we can solve for cos^2(75°):
-√3/2 = cos^2(75°) - (1 - cos^2(75°))
Simplifying further:
-√3/2 = 2cos^2(75°) - 1
Re-arranging the equation:
2cos^2(75°) = -√3/2 + 1
2cos^2(75°) = (-√3 + 2)/2
cos^2(75°) = (-√3 + 2)/4
Now, we substitute this value back into our main expression:
(2((-√3 + 2)/4) + 1 + (4/((-√3 + 2)/4))) /
(3((-√3 + 2)/4)^2 + 5((-√3 + 2)/4) + (4/((-√3 + 2)/4))^2)
Simplifying both the numerator and the denominator gives the final value of the expression.