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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 51
Chapter 2, Problem 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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Key Concepts

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

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
Related Practice
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