Final answer:
For angle theta in the fourth quadrant with tan theta equal to -3/5, the six trigonometric functions are sin theta = -3/√(34), cos theta = 5/√(34), tan theta = -3/5, cot theta = -5/3, sec theta = √(34)/5, and csc theta = -√(34)/3.
Step-by-step explanation:
Given that angle theta is in the fourth quadrant and tan theta is -3/5, we can calculate all the trigonometric functions for theta. In the fourth quadrant, cosine is positive, and sine is negative. Firstly, from the given tangent value, we can determine the opposite and adjacent sides relative to theta in a right triangle. If tan theta = -3/5, then the opposite side is -3 (since tangent is negative in the fourth quadrant) and the adjacent side is 5. To find the hypotenuse, we use the Pythagorean theorem:
hypotenuse = √(opposite^2 + adjacent^2) = √((-3)^2 + 5^2) = √(34)
Now, we can find the six trigonometric functions:
- sin theta = opposite/hypotenuse = -3/√(34)
- cos theta = adjacent/hypotenuse = 5/√(34)
- tan theta = opposite/adjacent = -3/5 (given)
- cot theta = 1/tan theta = -5/3
- sec theta = 1/cos theta = √(34)/5
- csc theta = 1/sin theta = -√(34)/3