Determine whether the positive or negative square root should be selected.
sin (-10°) = ± √[(1 - cos (-20°))/2]

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Recall the half-angle identity for sine: \(\sin \theta = \pm \sqrt{\frac{1 - \cos 2\theta}{2}}\). In this problem, \(\theta = -10^\circ\), so \(2\theta = -20^\circ\).

The expression given is \(\sin(-10^\circ) = \pm \sqrt{\frac{1 - \cos(-20^\circ)}{2}}\), which matches the half-angle formula. Our goal is to determine whether to use the positive or negative root.

Consider the sign of \(\sin(-10^\circ)\). Since sine is an odd function, \(\sin(-\alpha) = -\sin(\alpha)\), so \(\sin(-10^\circ) = -\sin(10^\circ)\).

Since \(\sin(10^\circ)\) is positive (because \(10^\circ\) is in the first quadrant where sine is positive), \(\sin(-10^\circ)\) must be negative.

Therefore, because \(\sin(-10^\circ)\) is negative, we select the negative square root in the half-angle formula: \(\sin(-10^\circ) = - \sqrt{\frac{1 - \cos(-20^\circ)}{2}}\).

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Half-Angle Identity for Sine

The half-angle identity expresses sine of an angle as ±√[(1 - cos(2θ))/2]. It helps find sin(θ) using the cosine of double the angle. The ± sign indicates that the sine value can be positive or negative depending on the angle's quadrant.

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Double Angle Identities

Sign of Sine Function in Different Quadrants

The sign of sine depends on the angle's quadrant on the unit circle. For negative angles, the position is measured clockwise from the positive x-axis. Since -10° lies in the fourth quadrant, where sine values are negative, the negative root should be chosen.

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Sum and Difference of Sine & Cosine

Relationship Between Negative Angles and Trigonometric Functions

Sine is an odd function, meaning sin(-θ) = -sin(θ). This property helps determine the sign of sine for negative angles by relating it to the positive angle's sine value, confirming the sign choice in the half-angle formula.

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