Let (theta) be an angle in quadrant II such that sec(theta) -4/3 .
Find the exact values of cot (theta) and sin (theta).
To find the values of cot (theta) and sin (theta), we need to use the information given about sec (theta).
1. We know that sec (theta) = 4/3.
Recall that sec (theta) is the reciprocal of cosine (theta), so we can rewrite the equation as:
cos (theta) = 3/4.
Now, we can use the Pythagorean Identity to find sin (theta):
sin^2 (theta) + cos^2 (theta) = 1.
Substituting the value of cos (theta) we found earlier, we get:
sin^2 (theta) + (3/4)^2 = 1.
Simplifying this equation, we have:
sin^2 (theta) + 9/16 = 1.
Moving the terms around, we get:
sin^2 (theta) = 1 - 9/16.
sin^2 (theta) = 16/16 - 9/16 = 7/16.
Now, we can take the square root of both sides to find sin (theta):
sin (theta) = ±√(7/16).
Since theta is in the second quadrant, sin (theta) is positive. Therefore:
sin (theta) = √(7/16).
To find cot (theta), we can use the fact that cot (theta) = cos (theta) / sin (theta).
Using the values we have:
cot (theta) = (3/4) / (√(7/16)).
To rationalize the denominator, we multiply the numerator and denominator by √16:
cot (theta) = (3/4) / (√(7/16)) * (√16/√16).
Simplifying, we get:
cot (theta) = (3√16) / (4√7).
Using rationalization of the denominator, we can simplify further:
cot (theta) = (3√(16*7)) / (4√(7*7)).
cot (theta) = (3√(112)) / (4√(49)).
Finally, we simplify to get the exact values of cot (theta) and sin (theta):
cot (theta) = (3√112) / (4√49) = (3√112) / (4*7) = (3√112) / 28.
sin (theta) = √(7/16) = √7/4.
Therefore, the exact values of cot (theta) and sin (theta) are (3√112)/28 and √7/4, respectively.