Final answer:
To find the values of the six trigonometric functions of theta when the terminal side lies on the line y=2x in Quadrant III, we need to determine the values of sine, cosine, tangent, cosecant, secant, and cotangent.
Step-by-step explanation:
To find the values of the six trigonometric functions of theta when the terminal side lies on the line y=2x in Quadrant III, we need to determine the values of sine, cosine, tangent, cosecant, secant, and cotangent.
In Quadrant III, both x and y coordinates are negative. Since the line y=2x has a positive slope, we can choose a point on the line, such as (-1, -2), to represent theta.
Using the properties of right triangles, we can calculate the values of the trigonometric functions:
- sine(theta) = y-coordinate/hypotenuse = -2/sqrt(-1^2 + (-2)^2)
- cosine(theta) = x-coordinate/hypotenuse = -1/sqrt(-1^2 + (-2)^2)
- tangent(theta) = sine(theta)/cosine(theta)
- cosecant(theta) = 1/sine(theta)
- secant(theta) = 1/cosine(theta)
- cotangent(theta) = 1/tangent(theta)