Final answer:
To find the trigonometric functions of theta where cot(theta) = -5/9 and cos(theta) > 0, we determined theta is in the fourth quadrant. Using the Pythagorean theorem, we calculated sin(theta) = -9/sqrt(106), cos(theta) = 5/sqrt(106), tan(theta) = -9/5, csc(theta) = sqrt(106)/-9, and sec(theta) = sqrt(106)/5.
Step-by-step explanation:
To solve for the trigonometric functions of θ given that cot(θ) = −5/9 and cos(θ) > 0, we must use the basic definitions and properties of trigonometric functions. Since cotangent is the reciprocal of tangent, we can express tangent as tan(θ) = −9/5. Knowing the signs of cotangent and cosine helps determine the quadrant in which the angle θ lies. In this case, since cot(θ) is negative and cos(θ) is positive, θ is in the fourth quadrant.
Using the Pythagorean identity, we can find the values of sin(θ) and cos(θ). Since tan(θ) = sin(θ)/cos(θ), and we have tan(θ) = −9/5, we can consider the opposite side to be −9, the adjacent side to be 5, and by the Pythagorean theorem, the hypotenuse would be √(9²+5²) = √106.
Now, we can determine the values of the trigonometric functions as follows:
- sin(θ) = −9/√106
- cos(θ) = 5/√106
- tan(θ) = −9/5
- csc(θ) = √106/−9 (since csc is the reciprocal of sin)
- sec(θ) = √106/5 (since sec is the reciprocal of cos)
Thus, we have found the values of the various trigonometric functions for the given angle θ based on the initial information provided.