Final answer:
Given that sin(Theta) = -4/7 and Theta is in Quadrant IV, we can find the other trigonometric functions by using the Pythagorean identity and considering the signs of the functions in the fourth quadrant. Cos(Theta) = √33/7, tan(Theta) = -4/√33, sec(Theta) = 7/√33, csc(Theta) = -7/4, and cot(Theta) = √33/-4.
Step-by-step explanation:
If sin(Theta) = -4/7 and Theta is in Quadrant IV, we can find the other trigonometric functions using the Pythagorean identity and the characteristics of the trigonometric functions in different quadrants.
In the fourth quadrant, cosine is positive and sine is negative. This means cos(Theta) will be positive. Since we know sin^2(Theta) + cos^2(Theta) = 1, we have (-4/7)^2 + cos^2(Theta) = 1. Solving for cos(Theta), we get cos(Theta) = √(1 - 16/49) = √(49/49 - 16/49) = √(33/49) = √33/7.
To find tan(Theta), we use the fact that tan(Theta) = sin(Theta)/cos(Theta). So, tan(Theta) = -4/7 / √33/7 = -4/√33.
For sec(Theta), which is 1/cos(Theta), we have sec(Theta) = 7/√33. To find csc(Theta), which is 1/sin(Theta), we have csc(Theta) = -7/4. Finally, cot(Theta) is the reciprocal of tan(Theta), so cot(Theta) = √33/-4.