Answer:
The given line is y = 1/4x, and the quadrant specified is quadrant III.
To find a point on the line that lies in quadrant III, we need to choose values for x and y that satisfy the equation y = 1/4x and are negative.
Let's choose x = -4 as an example. Plugging this into the equation, we have y = 1/4(-4) = -1.
So, the point (-4, -1) lies on the line in quadrant III.
Now, let's find the values of the six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) for the angle formed by the line and the positive x-axis.
First, let's calculate the length of the hypotenuse (r) using the Pythagorean theorem: r = sqrt((-4)^2 + (-1)^2) = sqrt(16 + 1) = sqrt(17).
Next, we can calculate the values of the trigonometric functions using the coordinates of the point (-4, -1):
1. Sine (sin): sin(theta) = y/r = -1/sqrt(17)
2. Cosine (cos): cos(theta) = x/r = -4/sqrt(17)
3. Tangent (tan): tan(theta) = y/x = -1/-4 = 1/4
4. Cosecant (csc): csc(theta) = 1/sin(theta) = -sqrt(17)
5. Secant (sec): sec(theta) = 1/cos(theta) = -sqrt(17)/4
6. Cotangent (cot): cot(theta) = 1/tan(theta) = 4
Therefore, the exact values of the six trigonometric functions for the angle formed by the line and the positive x-axis in quadrant III are:
1. sin(theta) = -1/sqrt(17)
2. cos(theta) = -4/sqrt(17)
3. tan(theta) = 1/4
4. csc(theta) = -sqrt(17)
5. sec(theta) = -sqrt(17)/4
6. cot(theta) = 4
Please note that there can be multiple points on the line in quadrant III, and this is just one example. The trigonometric function values may vary depending on the chosen point, but the ratios will remain the same
Explanation: