Find the values of the six trigonometric functions for an angle in standard position having each given point on its terminal side. Rationalize denominators when applicable. (6√3 , ―6)

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Identify the coordinates of the point on the terminal side of the angle: \(x = 6\sqrt{3}\) and \(y = -6\).

Calculate the radius (or hypotenuse) \(r\) using the distance formula: \(r = \sqrt{x^2 + y^2} = \sqrt{(6\sqrt{3})^2 + (-6)^2}\).

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

Substitute the values of \(x\), \(y\), and \(r\) into each function and simplify the expressions, rationalizing denominators where necessary.

Determine the signs of the trigonometric functions based on the quadrant in which the point lies (since \(x > 0\) and \(y < 0\), the point is in the fourth quadrant).

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Coordinates and the Terminal Side of an Angle

An angle in standard position has its vertex at the origin and its initial side along the positive x-axis. The given point (6√3, -6) lies on the terminal side of the angle, and its coordinates help determine the radius (distance from origin) and the signs of the trigonometric functions.

Definition of the Six Trigonometric Functions Using Coordinates

The six trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) can be defined using the coordinates (x, y) of a point on the terminal side and the radius r = √(x² + y²). Specifically, sin = y/r, cos = x/r, tan = y/x, and their reciprocals define the other three functions.

Rationalizing Denominators

When expressing trigonometric functions as fractions, denominators containing radicals should be rationalized for standard form. This involves multiplying numerator and denominator by a suitable radical to eliminate the root from the denominator, ensuring the expression is simplified and easier to interpret.