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Ch. 1 - Angles and the Trigonometric Functions
Blitzer - Trigonometry 3rd Edition
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Tutti i libri di testoBlitzer 3rd EditionCh. 1 - Angles and the Trigonometric FunctionsProblema 31
Capitolo 1, Problema 31

In Exercises 23–34, find the exact value of each of the remaining trigonometric functions of θ. tan θ = 4/3, cos θ < 0

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Identify the given information: \(\tan \theta = \frac{4}{3}\) and \(\cos \theta < 0\). Recall that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
Determine the quadrant of \(\theta\) based on the signs of \(\tan \theta\) and \(\cos \theta\). Since \(\tan \theta\) is positive and \(\cos \theta\) is negative, \(\theta\) lies in the second quadrant.
Use the identity \(\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{4}{3}\) to express \(\sin \theta\) and \(\cos \theta\) in terms of a common variable. Let \(\cos \theta = x\), then \(\sin \theta = \frac{4}{3} x\).
Apply the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) by substituting \(\sin \theta = \frac{4}{3} x\) and \(\cos \theta = x\), then solve for \(x\).
Once \(\cos \theta\) and \(\sin \theta\) are found, use these values to find the remaining trigonometric functions: \(\sec \theta = \frac{1}{\cos \theta}\), \(\csc \theta = \frac{1}{\sin \theta}\), and \(\cot \theta = \frac{1}{\tan \theta}\).

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Understanding Trigonometric Ratios

Trigonometric ratios relate the angles of a right triangle to the ratios of its sides. Given tan θ = 4/3, this means the opposite side over adjacent side is 4/3. Knowing one ratio allows calculation of the other trigonometric functions using definitions and relationships.
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Introduction to Trigonometric Functions

Sign of Trigonometric Functions in Quadrants

The sign of trigonometric functions depends on the quadrant where the angle lies. Since cos θ < 0, θ is in either the second or third quadrant. This information helps determine the correct signs of sine, cosine, and other functions when calculating their exact values.
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Quadratic Formula

Using the Pythagorean Identity

The Pythagorean identity, sin²θ + cos²θ = 1, allows finding missing trigonometric values when one function is known. After determining the sides from tan θ, this identity helps calculate sine and cosine values, ensuring the results satisfy the fundamental trigonometric relationship.
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Pythagorean Identities
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Domanda del libro di testo

In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of


0, 𝜋/4, 𝜋/2, 3𝜋/4, 𝜋, 5𝜋/4, 3𝜋/2, 7𝜋/4, and 2𝜋.


a. Use the (x,y) coordinates in the figure to find the value of the trigonometric function.

b. Use periodic properties and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.

cot 15𝜋/2

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In Exercises 31–38, find a cofunction with the same value as the given expression. sin 7°

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In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of


0, 𝜋/4, 𝜋/2, 3𝜋/4, 𝜋, 5𝜋/4, 3𝜋/2, 7𝜋/4, and 2𝜋.


a. Use the (x,y) coordinates in the figure to find the value of the trigonometric function.

b. Use periodic properties and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.

<IMAGE>


cot 𝜋/2

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In Exercises 29–34, convert each angle in degrees to radians. Round to two decimal places. -50°

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Find a cofunction with the same value as the given expression.

sin 19°

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In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of

0, 𝜋/4, 𝜋/2, 3𝜋/4, 𝜋, 5𝜋/4, 3𝜋/2, 7𝜋/4, and 2𝜋.

a. Use the (x,y) coordinates in the figure to find the value of the trigonometric function.

b. Use periodic properties and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.

<Image>

sin 47𝜋/4

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