SOLUTION: If (theta) is a second-quadrant with cos(theta)={{{2/sqrt5}}}, find the exact value of cos2(theta). a. -3/5 b. -2/5 c. 2/5 d. 3/5

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Question 73914This question is from textbook
: If (theta) is a second-quadrant with cos(theta)=, find the exact value of cos2(theta).
a. -3/5
b. -2/5
c. 2/5
d. 3/5
This question is from textbook

Found 2 solutions by stanbon, bucky:
Answer by stanbon(75887) (Show Source):
You can put this solution on YOUR website!
If (theta) is a second-quadrant with cos(theta)=, find the exact value of cos2(theta).
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cos(2theta)= cos^2(theta) - sin^2(theta)
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Need to find sin(theta)
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cos(theta) = adj/hyp = 2/sqrt5
adjacent is an x-value; it is negative in the 2nd quadrant;
therefore the cos(theta) is negative; it must be -2/sqrt5
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If adj = -2 and hyp = sqrt5 then opp = sqrt((sqrt5)^2-2^2))= 1
Therefore sin(theta)= opp/hyp = 1/2sqrt5
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Then cos(2theta)= (2/sqrt5)^2 - (1/sqrt5)^2 = 3/5
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Cheers,
Stan H.

Answer by bucky(2189) (Show Source):
You can put this solution on YOUR website!
This problem has an error in it. It tells you that theta is in the second quadrant,
but it gives you a positive value for the cosine of theta. You need to recognize that in
the second quadrant the cosine is negative. Cosine is only positive in quadrants I and IV.
.
Here's a way to do the problem quickly using a scientific calculator ($10 type).
.
Set you calculator into the degrees mode. Divide 2 by the square root of 5 and use the
change sign mode to make the answer a negative number. Your answer should be -0.894427191.
This is the decimal value of the cosine of theta.
.
Next use the arccosine function (cos^-1 function which is normally shift+cos keys) and you
should find that the angle whose cosine is -0.894427191 is 153.4349488 degrees. Multiply this
angle by 2 to get two theta. You should have 306.8698976 degrees for 2 theta. (Note that
this angle is in quadrant IV where the cosine is positive.) Now that you have found 2 theta,
just press the cosine key and you should find that the cosine of 2 theta is 0.6 which is the
same as +3/5.
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Answer d is the correct answer.
.
Hope this helps. Using a calculator makes it go fast.