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Ch. 1 - Angles and the Trigonometric Functions
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 1 - Angles and the Trigonometric FunctionsProblem 38
Chapter 1, Problem 38

In Exercises 37–38, a point on the terminal side of angle θ is given. Find the exact value of each of the six trigonometric functions of θ, or state that the function is undefined.
(0, -1)

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1
Identify the coordinates of the point on the terminal side of angle \( \theta \). Here, the point is \( (0, -1) \), so \( x = 0 \) and \( y = -1 \).
Calculate the radius \( r \), which is the distance from the origin to the point, using the formula \( r = \sqrt{x^2 + y^2} \). Substitute the values to get \( r = \sqrt{0^2 + (-1)^2} \).
Recall the definitions of the six trigonometric functions in terms of \( x \), \( y \), and \( r \): \[ \sin \theta = \frac{y}{r}, \quad \cos \theta = \frac{x}{r}, \quad \tan \theta = \frac{y}{x} \] \[ \csc \theta = \frac{r}{y}, \quad \sec \theta = \frac{r}{x}, \quad \cot \theta = \frac{x}{y} \]
Substitute the values of \( x \), \( y \), and \( r \) into each function. Note that if the denominator in any function is zero, that function is undefined.
Simplify each expression to find the exact values of the six trigonometric functions or state which are undefined based on the point \( (0, -1) \).

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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

The terminal side of an angle θ in standard position passes through a point (x, y) on the coordinate plane. This point helps determine the angle's trigonometric values by relating x and y to the radius r, which is the distance from the origin to the point.
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Definition of the Six Trigonometric Functions

The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—are defined using the coordinates (x, y) and radius r as follows: sin(θ) = y/r, cos(θ) = x/r, tan(θ) = y/x, and their reciprocals cosec(θ) = r/y, sec(θ) = r/x, cot(θ) = x/y. These definitions allow calculation of exact values.
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Handling Undefined Trigonometric Functions

Some trigonometric functions become undefined when their denominators are zero, such as tan(θ) = y/x when x = 0. Recognizing when a function is undefined is crucial, especially when the point lies on an axis, as in (0, -1), where division by zero may occur.
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Related Practice
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