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1. There are four positive integers a, b, c, and d such that for all values of x. Find a, b, c, and d.

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1. There are four positive integers a, b, c, and d such that $$4\cos(x)\cos(2x)\cos(4x) = \cos(ax) + \cos(bx) + \cos(cx) + \cos(dx)$$for all values of x. Find a, b, c, and d.

2. There are integers a, b, c, and d such that $$\tan (7.5^\circ) = \sqrt{a} + \sqrt{b} - \sqrt{c} - \sqrt{d}$$. Find a+b and c+d.

3. Find the value of $$\theta$$, $$0^\circ \le \theta \le 90^\circ$$, such that $$\cos( 5^\circ) = \sin (25^\circ) + \sin (\theta)$$ in degrees, not radians.

can someone help me? i've been stuck on these for quite a while for now and would appreciate some help!

Feb 8, 2020

#1
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1. You can write the expression as cos(2x) + cos(4x) + cos(6x) + cos(8x), so a + b + c + d = 2 + 4 + 6 + 8 = 20.

Feb 8, 2020
#2
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I'll try #!:

Using the formula:  cos(X) + cos(Y)  =  2·cos( (X + Y)/2 ) ·cos( (X -Y)/2 ):

Let  ax = 4x  and  bx  =  4x:  cos(ax) + cos(bx)  =  cos(4x) + cos(4x)   2 · cos( (4x + 4x)/2 ) · cos( (4x - 4x)/2 )

=  2 · cos( 8x/2 ) · cos( 0/2 )

=  2 · cos(4x) · cos(0)

=  2 · cos(4x) · 1

=  2 · cos(4x)

Let  cx = 3x  and  dx  =  1x:  cos(ax) + cos(bx)  =  cos(3x) + cos(x)   2 · cos( (3x + x)/2 ) · cos( (3x - x)/2 )

=  2 · cos( 4x/2 ) · cos( 2x/2 )

=  2 · cos(2x) · cos(x)

So we have:  2 · cos(4x) · 2 · cos(2x) · cos(x)  =  4 · cos(4x) · cos(2x) · cos(x)

Choose  cos(4x) + cos(4x) + cos(3x) + cos(1x)

Feb 9, 2020
#3
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In number 3, can you use a calculator?

If so:                               cos(5°)  =  sin(25°)  + sin(X)

cos(5°) - sin(25°)  =  sin(X)

Using inv(sin):

--->        =  35°

Feb 9, 2020
#4
+28449
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Here's my attempt at #2:

I should note that only the first result (ie a=2, b=6, c=3, d=4) matches tan(7.5°).

Feb 9, 2020
edited by Alan  Feb 9, 2020