+71 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!
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.
+23246 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)
