Question

Find the exact value of each expression.sin (13π/12)

Accepted Answer

Recognize that the angle $$ \frac{13\pi}{12}  is $$not one of the standard angles on the unit circle, so we need to express it as a sum or difference of angles whose sine values we know exactly. Rewrite $$ \frac{13\pi}{12}  as$$ $$ \pi + \frac{\pi}{12}  or as a $$sum of two angles such as $$ \frac{3\pi}{4} + \frac{\pi}{3}  or$$ $$ \pi - \frac{\pi}{12} . $$For this problem, use the sum $$ \frac{3\pi}{4} + \frac{\pi}{3} $$ because both $$ \frac{3\pi}{4} $$ and $$ \frac{\pi}{3} $$ are standard angles. Apply the sine addition formula: $$ \sin(a + b) = \sin a \cos b + \cos a \sin b . $$Here, $$ a = \frac{3\pi}{4} $$ and $$ b = \frac{\pi}{3} . $$Substitute the known exact values: $$ \sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2} $$, $$ \cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2} $$, $$ \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} $$, and $$ \cos \frac{\pi}{3} = \frac{1}{2} . $$Combine these values into the formula: $$ \sin \left( \frac{3\pi}{4} + \frac{\pi}{3} \right) = \sin \frac{3\pi}{4} \cos \frac{\pi}{3} + \cos \frac{3\pi}{4} \sin \frac{\pi}{3} $$, then simplify the expression step-by-step to find the exact value.

Find the exact value of each expression.
sin (13π/12)

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Unit Circle and Radian Measure

Angle Sum and Difference Identities

Exact Values of Sine and Cosine for Special Angles