In Exercises 1–8, a point on the terminal side of angle θ is given. Find the exact value of each of the six trigonometric functions of θ. (-1, -3)

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Identify the coordinates of the point on the terminal side of angle \( \theta \). Here, the point is \((-1, -3)\), so \(x = -1\) and \(y = -3\).

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 \( r = \sqrt{(-1)^2 + (-3)^2} \).

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

Substitute the values of \(x\), \(y\), and \(r\) into each of the six functions to express them exactly in terms of radicals and integers.

Simplify each expression if possible, and pay attention to the signs of the functions based on the quadrant where the point \((-1, -3)\) lies (which is the third quadrant).

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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). 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.

Radius (r) and Its Calculation

The radius r is the distance from the origin to the point (x, y) on the terminal side, calculated using the Pythagorean theorem: r = √(x² + y²). It is essential for normalizing the coordinates when finding trigonometric function values.

Six Trigonometric Functions and Their Definitions

The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—are defined using x, y, and r: sin θ = y/r, cos θ = x/r, tan θ = y/x, and their reciprocals csc θ, sec θ, cot θ. Knowing these definitions allows exact value computation from coordinates.