Final answer:
To sketch an angle θ with the point (-5,12) on the terminal side, determine the quadrant, and use the trigonometric functions to find the exact values.
Step-by-step explanation:
To sketch an angle θ in standard position such that the point (-5,12) is on the terminal side of θ with the least possible positive measure, we need to determine the quadrant in which the point (-5,12) lies.
In the second quadrant, the x-coordinate is negative and the y-coordinate is positive. Therefore, if the point (-5,12) is on the terminal side of the angle, θ must be between 90 and 180 degrees.
To find the exact values of the six trigonometric functions for θ, we can use the information given.
The sine function is equal to the ratio of the y-coordinate (12) to the hypotenuse (the distance from the origin to the point), which is the square root of the sum of the squares of the x-coordinate (-5) and the y-coordinate (12).
Similarly, the cosine function is equal to the ratio of the x-coordinate (-5) to the hypotenuse.
The tangent function is equal to the ratio of the sine of θ to the cosine of θ.
The cosecant function is equal to the reciprocal of the sine of θ: 1/sin(θ).
The secant function is equal to the reciprocal of the cosine of θ: 1/cos(θ).
The cotangent function is equal to the reciprocal of the tangent of θ: 1/tan(θ).