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. (3 , ―4)

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Identify the coordinates of the point on the terminal side of the angle: \((x, y) = (3, -4)\).

Calculate the radius (or hypotenuse) \(r\) using the distance formula: \(r = \sqrt{x^2 + y^2} = \sqrt{3^2 + (-4)^2}\).

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 to express them as fractions.

Rationalize the denominators where necessary by multiplying numerator and denominator by the appropriate conjugate or factor.

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주요 개념

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Trigonometric Functions and the Coordinate Plane

Trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) can be defined using coordinates of a point on the terminal side of an angle in standard position. The x- and y-coordinates correspond to the adjacent and opposite sides of a right triangle, while the distance from the origin is the hypotenuse.

Distance Formula and Hypotenuse Calculation

To find the hypotenuse (r) for the point (x, y), use the distance formula r = √(x² + y²). This value is essential for calculating the trigonometric functions, as sine and cosine depend on y/r and x/r respectively, and the other functions are derived from these ratios.

Rationalizing Denominators

When expressing trigonometric functions as fractions, denominators containing square roots should be rationalized. This involves multiplying numerator and denominator by the radical to eliminate the root from the denominator, resulting in a simplified and standardized form.