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find the exact values of the expressions without solving for x.

If sin(x) = 1/4 and x is in quadrantI,find the exact values of the expressions without solving for x.
 
(a) sin(2x)
(b) cos(2x)
(c) tan(2x)

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cos 2x = 2cos2x - 1
sin 2x = 2sinxcosx
tan2x = (2tanx)/(1-tan2x)
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2 Answers

Hi again, Lauren.
 
A triangle is worth a zillion words...
 
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/      |     Use your imagination...  :-)
 
 
Remember that sin(x) = opposite side/hypotenuse.  So on our triangle, we are going to put the angle x in the bottom left vertex.  That makes the vertical leg the opposite leg = 1 and the hypotenuse = 4
 
  
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Now use the Pythag Thm to determine that the horizontal leg √15.  So the cos(x)=(√15)/4
 
If you are not to solve for "x", can you use the trig double angle formulas?
 
a) sin(2x) = 2sin(x)cos(x) = 2(1/4)(√15)/4 = (1/2)(√15)/4 = (√15)/8
 
b) cos(2x) = cos2(x) - sin2(x) = [(√15/4)(√15/4)] - (1/4)(1/4) = 15/16 - 1/16 = 14/16 = 7/8
 
c) tan(2x) = [sin(2x)/cos(2x)] = [√15/8]/[7/8] = (√15)/7
 
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Use the Pythagorean Theorem to find the length of the adjacent side in the right triangle in Quad I.
a2 + 12 = 42
a2 = 15, a = √15
cos x = (√15)/4    tan x = 1/√15 = √15/15
Now use these formulas: