Question

Use a half-angle identity to find each exact value.sin 195°

Accepted Answer

Recognize that 195° is not a standard angle, but it can be expressed in terms of an angle whose sine or cosine is known. Notice that 195° = 2 × 97.5°, so we can use the half-angle identity for sine by setting $$ 	heta = 195° $$ and $$ \frac{	heta}{2} = 97.5° . $$However, since 97.5° is not a standard angle either, let's try to express 195° as $$ 180° + 15°  to $$use angle sum identities or consider the half-angle identity for $$ 195° = 2 	imes 97.5° . $$Alternatively, use the half-angle identity with $$ 	heta = 390° $$ because $$ 195° = \frac{390°}{2} . $$Recall the half-angle identity for sine:

$$ \sin\left(\frac{\alpha}{2}ight) = \pm \sqrt{\frac{1 - \cos(\alpha)}{2}} $$

Here, $$ \alpha = 390°  so $$that $$ \sin(195°) = \sin\left(\frac{390°}{2}ight) . $$Determine the sign of $$ \sin(195°) . $$Since 195° is in the third quadrant (between 180° and 270°), and sine is negative in the third quadrant, the sign will be negative. Calculate $$ \cos(390°) . $$Since 390° is coterminal with 30° (because 390° - 360° = 30°), $$ \cos(390°) = \cos(30°) . $$Use the known exact value $$ \cos(30°) = \frac{\sqrt{3}}{2} . $$Substitute $$ \cos(390°) = \frac{\sqrt{3}}{2} $$ into the half-angle formula and include the negative sign determined earlier:

$$ \sin(195°) = - \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} $$

This expression represents the exact value of $$ \sin(195°) .$$

Use a half-angle identity to find each exact value.
sin 195°

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Key Concepts

Half-Angle Identities

Reference Angles and Quadrants

Exact Values of Cosine for Special Angles