Answer:
Since the point (-3, 2) lies on the terminal side of the angle theta in standard position, we can use the Pythagorean theorem to find the length of the hypotenuse:
h = sqrt((-3)^2 + 2^2) = sqrt(9 + 4) = sqrt(13)
Then we can use the definitions of sin and cot:
sin(theta) = opposite/hypotenuse = 2/sqrt(13)
cot(theta) = adjacent/opposite = -3/2
The length of the arc intercepted by a central angle of pi/16 radians in a circle of diameter 24 feet is given by:
L = r * theta
where r is the radius of the circle, which is half the diameter: r = 12 feet. So we have:
L = 12 * (pi/16) = 3pi/4 feet
To approximate to the nearest hundredth, we can use 3.14 for pi:
L ≈ 2.36 feet
So the length of the arc intercepted by the central angle is approximately 2.36 feet.
We can use the law of sines to solve for the remaining angle and side lengths of the triangle:
a/sin(A) = b/sin(B) = c/sin(C)
where a, b, c are the side lengths and A, B, C are the opposite angles. We are given a = 9 and b = 10, and A = 42°. We can solve for sin(B) using the equation above:
10/sin(B) = 9/sin(42°)
sin(B) = 10*sin(42°)/9 ≈ 0.742
This means that angle B is acute, and we can solve for it using the inverse sine function:
B = sin^(-1)(0.742) ≈ 47.7°
Now we can find the remaining angle C by subtracting A and B from 180°:
C = 180° - A - B ≈ 90.3°
Finally, we can use the law of sines again to find the remaining side length c:
c/sin(C) = a/sin(A)
c/sin(90.3°) = 9/sin(42°)
c ≈ 11.1
So the solution for the triangle is: a = 9, b = 10, c ≈ 11.1, A = 42°, B ≈ 47.7°, C ≈ 90.3°.
Explanation: