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Ch. 1 - Trigonometric Functions
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 1 - Trigonometric FunctionsProblem 31
Chapter 2, Problem 31

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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1
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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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.
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Related Practice
Textbook Question

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Textbook Question

Sketch an angle θ in standard position such that θ has the least positive measure, and the given point is on the terminal side of θ. Then find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. See Examples 1, 2, and 4. (―2√3 , 2)

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Determine the signs of the trigonometric functions of an angle in standard position with the given measure. See Example 2.

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