Answers:
x is in quadrant I (first quadrant) in the upper right hand corner
sin(x) = sqrt(3)/2
tan(x) = sqrt(3)
where 'sqrt' is shorthand for 'square root'
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Explanation:
cos(x) = 1/2 shows us that cos(x) > 0. At the same time, we're told that tan(x) > 0. Both cosine and tangent are positive. This only happens when we're in the first quadrant. The first quadrant is to the right of the vertical y axis, and it is above the horizontal x axis. In short, the first quadrant is in the upper right hand corner. Tangent is positive in quadrant III; however, cosine is negative here. Similarly, cosine is positive in Q4, but tangent is negative here.
Use the pythagorean trig identity to determine the value of sin(x)
sin^2(x) + cos^2(x) = 1
sin^2(x) + (1/2)^2 = 1
sin^2(x) + 1/4 = 1
sin^2(x) = 1 - 1/4
sin^2(x) = 3/4
sin(x) = sqrt(3/4)
sin(x) = sqrt(3)/sqrt(4)
sin(x) = sqrt(3)/2 ... sine is positive in quadrant I
Now divide the values of sine and cosine to get tangent
tan(x) = sin(x)/cos(x)
tan(x) = sin(x) divided by cos(x)
tan(x) = sqrt(3)/2 divided by 1/2
tan(x) = sqrt(3)/2 times 2/1
tan(x) = sqrt(3) .... note how the '2's cancel
we see that tangent is positive, so that helps confirm the answer.
