Ch. 5 - Trigonometric Identities

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. 5 - Trigonometric Identities

Problema 5.RE.38

Capitolo 6, Problema 5.RE.38

Use the given information to find each of the following.
sin A/2, given cos A/2 = - 3, 90° < A < 180°

Guida verificata passo dopo passo

First, recognize that the problem gives you \( \cos \frac{A}{2} = -3 \) and the angle \( A \) is between 90° and 180°. Since \( \cos \frac{A}{2} \) must be between -1 and 1 for real angles, check if the given value is valid or if there might be a typo or misunderstanding in the problem statement.

Assuming the value is valid or corrected, recall the Pythagorean identity for sine and cosine: \( \sin^2 \theta + \cos^2 \theta = 1 \). Here, \( \theta = \frac{A}{2} \). Use this to express \( \sin \frac{A}{2} \) in terms of \( \cos \frac{A}{2} \):

\[ \sin \frac{A}{2} = \pm \sqrt{1 - \cos^2 \frac{A}{2}} \]

Determine the correct sign (positive or negative) for \( \sin \frac{A}{2} \) by considering the quadrant in which \( \frac{A}{2} \) lies. Since \( 90^\circ < A < 180^\circ \), then \( 45^\circ < \frac{A}{2} < 90^\circ \), which places \( \frac{A}{2} \) in the first quadrant where sine is positive.

Finally, substitute the value of \( \cos \frac{A}{2} \) into the formula and simplify under the square root to find \( \sin \frac{A}{2} \). Remember to choose the positive root based on the quadrant analysis.

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Half-Angle Identities

Half-angle identities relate the trigonometric functions of half an angle to those of the original angle. For sine and cosine, these identities help find values like sin(A/2) or cos(A/2) using known values of cos(A) or sin(A). They are essential for solving problems involving angles divided by two.

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Percorso guidato

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Double Angle Identities

Sign Determination in Quadrants

The sign of trigonometric functions depends on the quadrant in which the angle lies. Since A is between 90° and 180°, A/2 lies between 45° and 90°, placing it in the first quadrant where sine is positive and cosine is positive. This helps determine the correct sign of sin(A/2) given cos(A/2).

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Quadratic Formula

Pythagorean Identity

The Pythagorean identity states that sin²θ + cos²θ = 1 for any angle θ. This identity allows calculation of one trigonometric function if the other is known. In this problem, knowing cos(A/2) enables finding sin(A/2) by rearranging the identity.

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Percorso guidato

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Pythagorean Identities

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