In Exercises 37–38, a point on the terminal side of angle θ is given. Find the exact value of each of the six trigonometric functions of θ, or state that the function is undefined.
(0, -1)

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Identify the coordinates of the point on the terminal side of angle \( \theta \). Here, the point is \( (0, -1) \), so \( x = 0 \) and \( y = -1 \).

Calculate the radius \( r \), which is the distance from the origin to the point, using the formula \( r = \sqrt{x^2 + y^2} \). Substitute the values to get \( r = \sqrt{0^2 + (-1)^2} \).

Recall the definitions of the six trigonometric functions in terms of \( x \), \( y \), and \( r \): \[ \sin \theta = \frac{y}{r}, \quad \cos \theta = \frac{x}{r}, \quad \tan \theta = \frac{y}{x} \] \[ \csc \theta = \frac{r}{y}, \quad \sec \theta = \frac{r}{x}, \quad \cot \theta = \frac{x}{y} \]

Substitute the values of \( x \), \( y \), and \( r \) into each function. Note that if the denominator in any function is zero, that function is undefined.

Simplify each expression to find the exact values of the six trigonometric functions or state which are undefined based on the point \( (0, -1) \).

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Coordinates and the Terminal Side of an Angle

The terminal side of an angle θ in standard position passes through a point (x, y) on the coordinate plane. This point helps determine the angle's trigonometric values by relating x and y to the radius r, which is the distance from the origin to the point.

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Handling Undefined Trigonometric Functions

Some trigonometric functions become undefined when their denominators are zero, such as tan(θ) = y/x when x = 0. Recognizing when a function is undefined is crucial, especially when the point lies on an axis, as in (0, -1), where division by zero may occur.

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