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Ch. 1 - Angles and the Trigonometric Functions
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 3rd EditionCh. 1 - Angles and the Trigonometric FunctionsProblema 10
Capitolo 1, Problema 10

In Exercises 9–16, evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. tan 𝜋

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Recognize that the angle given is a quadrantal angle, specifically \(\pi\) radians, which corresponds to 180 degrees on the unit circle.
Recall that the tangent function is defined as the ratio of sine to cosine: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
Evaluate \(\sin \pi\) and \(\cos \pi\) using the unit circle values: \(\sin \pi = 0\) and \(\cos \pi = -1\).
Substitute these values into the tangent formula: \(\tan \pi = \frac{0}{-1}\).
Simplify the fraction to find the value of \(\tan \pi\), noting that division by a nonzero number is defined.

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Quadrantal Angles

Quadrantal angles are angles that lie on the x- or y-axis in the coordinate plane, typically multiples of 90° or π/2 radians. These angles include 0, π/2, π, 3π/2, and 2π, where trigonometric functions often take special values or become undefined.
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Quadratic Formula

Tangent Function at Quadrantal Angles

The tangent function is defined as the ratio of sine to cosine (tan θ = sin θ / cos θ). At quadrantal angles, since cosine or sine can be zero, the tangent may be zero, a finite number, or undefined if division by zero occurs.
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Evaluating Trigonometric Functions Using the Unit Circle

The unit circle provides coordinates (cos θ, sin θ) for any angle θ. Evaluating trigonometric functions at quadrantal angles involves identifying these coordinates and applying definitions, which helps determine exact values or identify undefined expressions.
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Evaluate Composite Functions - Values Not on Unit Circle
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In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of 0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋. 6 3 2 3 6 6 3 2 3 6 Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.

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Use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.


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sec 45°

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