Let (theta) be an angle in quadrant II such that csc(theta)=8/5 Find the exact value of tan(theta) and cos(theta).
To find the exact value of tan(theta) and cos(theta) when csc(theta) is given, we need to first determine the value of sin(theta) using the reciprocal relationship between csc(theta) and sin(theta).
Recall that the reciprocal of csc(theta) is sin(theta). Therefore, we have sin(theta) = 1/csc(theta).
Since csc(theta) = 8/5, we can substitute this value in the equation to find sin(theta):
sin(theta) = 1 / (8/5)
sin(theta) = 5/8
Now, to find the exact value of tan(theta), we can use the relationship between sin(theta), cos(theta), and tan(theta) in quadrant II. For an angle in quadrant II, sin(theta) is positive, while cos(theta) is negative.
We know that tan(theta) = sin(theta) / cos(theta)
Given that sin(theta) = 5/8, we can find cos(theta) by using the Pythagorean identity:
cos(theta) = sqrt(1 - sin^2(theta))
cos(theta) = sqrt(1 - (5/8)^2)
cos(theta) = sqrt(1 - 25/64)
cos(theta) = sqrt(64/64 - 25/64)
cos(theta) = sqrt(39/64)
So, the exact value of cos(theta) is sqrt(39/64).
Therefore, the exact value of tan(theta) is (5/8) / sqrt(39/64), which simplifies to:
tan(theta) = (5/8) * (sqrt(64)/sqrt(39))
tan(theta) = (5/8) * (8/sqrt(39))
tan(theta) = 5/sqrt(39)
Hence, the exact values of tan(theta) and cos(theta) are 5/sqrt(39) and sqrt(39/64), respectively.