Final answer:
The point (-16, 12) suggests the triangle is in the second quadrant. Using the distance to the point as the hypotenuse, the trigonometric functions are found: sin(θ) = 3/5, cos(θ) = -4/5, tan(θ) = -3/4, and cot(θ) = -4/3.
Step-by-step explanation:
To find the exact values for the trigonometric functions of θ given the point (-16, 12), we need to use the definitions of sine, cosine, and tangent with respect to a right triangle. For the given point, let's consider a right triangle formed with the point, the origin, and the point's projection on the x-axis. In this scenario, the x-coordinate (-16) represents the adjacent side, the y-coordinate (12) represents the opposite side, and the distance from the origin to the point, which can be calculated using the Pythagorean theorem, serves as the hypotenuse.
First, we calculate the hypotenuse, h:
h = √((-16)2 + 122)
h = √(256 + 144)
h = √(400)
h = 20
Next, we use the hypotenuse to find the trigonometric functions:
- sin(θ) = opposite/hypotenuse = 12/20 = 3/5
- cos(θ) = adjacent/hypotenuse = -16/20 = -4/5
- tan(θ) = opposite/adjacent = 12/(-16) = -3/4
- cot(θ) = 1/tan(θ) = 1/(-3/4) = -4/3
The signs of sine and cosine indicate that θ is in the second quadrant, where sine is positive and cosine is negative.