Question

Take an angle θ and position it in a coordinate system so that its vertex is at the origin and one of its sides is on the positive x-axis. If P(-3,4) is a point on the second side of the angle, what is the value of sin θ + cot θ?

Answer 1

Okay, let's break this down step-by-step:

• We are given an angle θ with its vertex at the origin (0, 0)
• One side of the angle is on the positive x-axis
• The point P(-3, 4) lies on the second side of the angle
• To find the angle θ, we use the slope formula: slope = (y2 - y1) / (x2 - x1)
• where (x1, y1) and (x2, y2) are the endpoints of the side
• Here: (x1, y1) = (0, 0) (origin) and (x2, y2) = (-3, 4) (point P)
• So slope = (4 - 0) / (-3 - 0) = 4 / -3 = -4/3
• And the angle has an inverse sine: θ = sin^-1(-4/3)

θ = - sin^-1(4/3)

θ = -1.18 radians

Now we have the value of θ, -1.18 radians.

• sin θ = sin(-1.18) = -0.8
• cot θ = cos(-1.18) / sin(-1.18) = -0.9104

Therefore:

sin θ + cot θ = -0.8 + -0.9104 = -1.7104

So the final answer is:

-1.7104

Does this make sense? Let me know if you need more details!