Final answer:
The given sin(θ) = 5/2 is incorrect as sine values must be between -1 and 1. Assuming a valid sine value, to find the remaining trig functions, one would use the Pythagorean theorem to determine the other sides of the triangle and calculate the values. The angle lies in either the second or fourth quadrant based on the given negative tangent.
Step-by-step explanation:
The student has provided sin(θ) = 5/2, which actually is not possible since the sine function has a range from -1 to 1. This might be a typographical error, and the sine value should be within the range -1 to 1. However, if we proceed with the concept, assuming that a correct sine value has been given, and considering that tan(θ) < 0, it indicates that the angle θ lies in either the second or fourth quadrant of the unit circle (since tan is positive in the first and third quadrants).
To find the remaining trig functions, one would typically use the given sine value to find 'y' and the Pythagorean theorem to find 'x' (given a unit circle where the hypotenuse is 1, or the hypotenuse 'h' in a right triangle). Then, use the x and y values to find cosine (x/h or x), tangent (y/x), cosecant (1/sin), secant (1/cos), and cotangent (x/y).
However, since the given sine value is invalid, it is not possible to accurately complete the problem as stated. If the sine value were within the correct range, one could use the following definitions based on a right triangle with sides x (adjacent), y (opposite), and h (hypotenuse):
- Cosine: x/h or cos(θ)
- Tangent: y/x or tan(θ)
- Cosecant: h/y or csc(θ)
- Secant: h/x or sec(θ)
- Cotangent: x/y or cot(θ)