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. (5, -5)

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Identify the coordinates of the point on the terminal side of angle \( \theta \). Here, the point is \( (5, -5) \), so \( x = 5 \) and \( y = -5 \).

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 \( r = \sqrt{5^2 + (-5)^2} \).

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, keeping in mind the signs of \( x \), \( y \), and \( r \) to determine the correct sign of each trigonometric function.

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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, calculated using the Pythagorean theorem.

Definition of the Six Trigonometric Functions

The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—are ratios involving x, y, and r. Specifically, sin(θ) = y/r, cos(θ) = x/r, tan(θ) = y/x, and their reciprocals define the other three functions. These ratios are essential for finding exact values.

Sign of Trigonometric Functions Based on Quadrants

The signs of x, y, and r determine the sign of each trigonometric function. Since r is always positive, the signs depend on the quadrant where the point lies. For (5, -5), the point is in the fourth quadrant, affecting the positivity or negativity of sine, cosine, and tangent values.