SOLUTION: if cos theta = 1/3, theta is in Q IV, find the exact value of sin(theta+pie/3)

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Question 919721: if cos theta = 1/3, theta is in Q IV, find the exact value of sin(theta+pie/3)
Found 2 solutions by ewatrrr, stanbon:
Answer by ewatrrr(24785) (Show Source):
You can put this solution on YOUR website!

Hi
cos(θ) = 1/3 Then sin(θ)= √2/3 Note: 1^2 + (√2)^2 = 3
sin(θ + π/3) = sinθcos(π/3) + cosθsin(π/3)
sin(θ + π/3) = (√2/3)cos(π/3) + (1/3)sin(π/3)
Plug and Play
sin(θ + π/3) = (√2/3)(1/2) + (1/3)(√3/2)  
............
θ radians sin θ cos θ tan θ
0� 0 0 1 0
30� π/6 1/2 √3/2 √3/3
45� π/4 √2/2 √2/2 1
60� π/3 √3/2 1/2 √3
90� π/2 1 0 ─
    

Answer by stanbon(75887) (Show Source):
You can put this solution on YOUR website!
if cos theta = 1/3, theta is in Q IV, find the exact value of sin(theta+pie/3)
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By definition, cos = x/r
So x = 1 and r = 3
Then y = -sqrt(3^2-1^2] = -sqrt(8)
Therefore
sin(t) = -sqrt(8)/3 = -(2/3)sqrt(2)
and cos(t) = 1/3
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Ans:: sin(t + pi/3) = sin(t)cos(pi/3)+cos(t)sin(pi/3)
= -(2/3)sqrt(2)*(1/2) + (1/3)(sqrt(3)/2)
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= -sqrt(2)/3 + sqrt(3)/6
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= [sqrt(3) - 2sqrt(2)]/6
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Cheers,
Stan H.
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