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Ch. 1 - Trigonometric Functions
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 1 - Trigonometric FunctionsProblema 51
Capitolo 2, Problema 51

An equation of the terminal side of an angle θ in standard position is given with a restriction on x. Sketch the least positive such angle θ , and find the values of the six trigonometric functions of θ . See Example 3. 2x + y = 0 , x ≥ 0

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1
Rewrite the given equation of the terminal side in slope-intercept form to understand the line better. Starting with the equation \(2x + y = 0\), solve for \(y\) to get \(y = -2x\).
Since the terminal side lies on the line \(y = -2x\) with the restriction \(x \geq 0\), consider a point on this line where \(x\) is positive. For simplicity, choose \(x = 1\), then \(y = -2(1) = -2\). This point \((1, -2)\) lies on the terminal side.
Determine the angle \(\theta\) in standard position whose terminal side passes through the point \((1, -2)\). Use the definition of tangent: \(\tan(\theta) = \frac{y}{x} = \frac{-2}{1} = -2\). Since \(x \geq 0\) and \(y < 0\), the point is in the fourth quadrant, so \(\theta\) is the least positive angle between \(0\) and \(2\pi\) with \(\tan(\theta) = -2\).
Calculate the six trigonometric functions of \(\theta\) using the coordinates of the point \((1, -2)\). First, find the radius \(r = \sqrt{x^2 + y^2} = \sqrt{1^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}\). Then, use the definitions: \(\sin(\theta) = \frac{y}{r}\), \(\cos(\theta) = \frac{x}{r}\), \(\tan(\theta) = \frac{y}{x}\), \(\csc(\theta) = \frac{r}{y}\), \(\sec(\theta) = \frac{r}{x}\), and \(\cot(\theta) = \frac{x}{y}\).
Summarize the values of the six trigonometric functions based on the point and radius found, keeping in mind the signs of each function in the fourth quadrant.

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Standard Position of an Angle

An angle in standard position has its vertex at the origin and its initial side along the positive x-axis. The terminal side is determined by rotating the initial side counterclockwise by the angle θ. Understanding this helps in visualizing and sketching the angle based on given line equations.
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Drawing Angles in Standard Position

Equation of a Line and Angle Determination

The terminal side of the angle lies along a line given by an equation, here 2x + y = 0. By rewriting the line in slope-intercept form, the slope corresponds to the tangent of the angle θ. The restriction x ≥ 0 limits the terminal side to the right half-plane, ensuring the least positive angle is found.
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Parameterizing Equations

Six Trigonometric Functions

The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—are ratios of the sides of a right triangle or coordinates on the unit circle. Once θ is identified, these functions can be calculated using the coordinates of a point on the terminal side or the slope of the line.
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Introduction to Trigonometric Functions
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