Question

Find all six trigonometric functions of theta if the given point is on the terminal side of theta. (If an answer is undefined, enter UNDEFINED.) (-2, -3)

sin θ = ____
cos θ = _____________
tan θ = __________
csc θ = _________
sec θ = _______
cot θ = ______

Answer 1

To find the trigonometric functions of θ for the point (-2, -3), we calculate the hypotenuse of the triangle formed by this point and the origin, and then use it to find sin θ, cos θ, tan θ, csc θ, sec θ, and cot θ.

Step-by-step explanation:

Given the point (-2, -3) we can find all trigonometric functions of θ. First, we determine the hypotenuse using the Pythagorean theorem: hypotenuse = √((-2)2 + (-3)2) = √(4+9) = √13. Now we can find each function:

• sin θ = opposite/hypotenuse = (-3)/√13
• cos θ = adjacent/hypotenuse = (-2)/√13
• tan θ = opposite/adjacent = (-3)/(-2) = 3/2
• csc θ = 1/sin θ = √13/(-3)
• sec θ = 1/cos θ = √13/(-2)
• cot θ = 1/tan θ = (-2)/(-3) = 2/3

Remember to take into account the negative signs given the position of the point in the coordinate plane, which affects the signs of the trigonometric functions.