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

검증된 단계별 안내

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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주요 개념

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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.

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).

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.

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