Question

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

Accepted Answer

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

$$ 
\cos \left( \frac{	heta}{2} ight) = \pm \sqrt{ \frac{1 + \cos 	heta}{2} } 
$$

Since we want $$ \cos 195° $$, we can rewrite it as $$ \cos (2 	imes 97.5°) $$ and then use the double-angle formula or use the half-angle identity by setting $$ 	heta = 390°  ($$which is $$ 2 	imes 195° ) $$and then find $$ \cos 195° = \cos \left( \frac{	heta}{2} ight) . $$But this might be complicated, so instead, let's use the half-angle identity by expressing 195° as $$ 180° + 15° $$ and then use the cosine addition formula or the half-angle identity for 15°. Alternatively, express 195° as $$ 180° + 15° . $$Since cosine has the property:

$$ 
\cos (180° + x) = -\cos x

we $$can write:

$$ 
\cos 195° = \cos (180° + 15°) = -\cos 15° 
$$

Now, focus on finding $$ \cos 15° $$ using the half-angle identity. To find $$ \cos 15° $$, use the half-angle identity with $$ 	heta = 30° $$ because $$ 15° = \frac{30°}{2} . $$Apply the half-angle formula:

$$ 
\cos 15° = \cos \left( \frac{30°}{2} ight) = \pm \sqrt{ \frac{1 + \cos 30°}{2} } 
$$

Determine the sign by considering the quadrant of 15°, which is in the first quadrant where cosine is positive. Substitute the known value $$ \cos 30° = \frac{\sqrt{3}}{2} $$ into the formula:

$$ 
\cos 15° = \sqrt{ \frac{1 + \frac{\sqrt{3}}{2}}{2} } 
$$

Simplify the expression inside the square root to get the exact value of $$ \cos 15° . $$Finally, recall from step 3 that $$ \cos 195° = -\cos 15° $$, so multiply your result by -1 to find the exact value of $$ \cos 195° .$$

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

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Half-Angle Identities

Reference Angles and Quadrants

Exact Values of Cosine for Common Angles