Final answer:
To determine the signs of the six trigonometric functions for a specific angle, you must consider the quadrant in which the angle lies. 'All Students Take Calculus' (ASTC) is a useful mnemonic to remember which functions are positive in each quadrant: all in the first, sine and cosecant in the second, tangent and cotangent in the third, and cosine and secant in the fourth.
Step-by-step explanation:
To find the signs of the six trigonometric function values for a given angle, you must consider the angle's location within the coordinate system. The six trigonometric functions are sine (θ), cosine (θ), tangent (θ), cosecant (θ), secant (θ), and cotangent (θ). Each function's sign depends on which quadrant the angle lies in after being positioned in standard position (with its vertex at the origin and its initial side along the positive x-axis).
In the first quadrant, all six functions are positive. In the second quadrant, sine and cosecant are positive, while the others are negative. In the third quadrant, tangent and cotangent are positive, and in the fourth quadrant, cosine and secant are positive. To remember this, the acronym 'ASTC' can be used, standing for All Students Take Calculus, which corresponds to All functions positive in the first quadrant, only Sine and its reciprocal function Cosecant positive in the second, Tangent and Cotangent positive in the third, and Cosine and its reciprocal Secant positive in the fourth quadrant.