Final Answer:
If P(x,y) is a point on the unit circle corresponding to t, then the equation that does not define a trigonometric function is c) cot t = y/x, x ≠ 0.
Step-by-step explanation:
To determine which of the options does not accurately define a trigonometric function, let's review each of the given equations in the context of the unit circle and the standard definitions of the trigonometric functions:
A. csc t = 1/y, y ≠ 0
The cosecant (csc) of an angle t in the unit circle is defined as the reciprocal of the sine (sin) of t. Since sine is given by sin t = y for a point P(x, y) on the unit circle, the cosecant is indeed csc t = 1/sin t = 1/y, provided that y ≠ 0 (since you cannot divide by zero). Therefore, option A is correct.
B. cos t = x
The cosine (cos) of an angle t in the unit circle is defined as the x-coordinate of the point P(x, y) on the unit circle. Thus, cos t = x correctly defines the cosine function. Option B is correct.
C. cot t = y/x, x ≠ 0
The cotangent (cot) of an angle t in the unit circle is defined as the reciprocal of the tangent (tan) of t. The tangent is given by tan t = y/x (where P(x, y) is the point on the unit circle and x ≠ 0 to avoid division by zero). However, this implies that the cotangent should be the reciprocal of this, i.e., cot t = x/y, not y/x as stated in option C. Therefore, option C incorrectly defines the cotangent function and is the answer we're looking for.
D. sec t = 1/x, x ≠ 0
The secant (sec) function is the reciprocal of the cosine function. Since cosine is given by cos t = x, the secant will be sec t = 1/cos t = 1/x, assuming x ≠ 0 because you cannot divide by zero. Therefore, option D correctly defines the secant function.
In conclusion, option C is incorrect as it does not accurately define the cotangent trigonometric function. The correct definition should be cot t = x/y, not y/x.