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Question

Find the exact value of the expression.

sin (-15 degrees)

Solution

Verified
Answered 2 years ago
Answered 2 years ago
Step 1
1 of 2

Notice that 15°=45°60°-15\text{\textdegree}=45\text{\textdegree}-60\text{\textdegree}, so

sin(15°)=sin(45°60°)\sin (-15\text{\textdegree})=\sin(45\text{\textdegree}-60\text{\textdegree})

Apply formula (2):

sin(45°60°)==sin45°cos60°cos45°sin60°==22122232==2464==264\sin(45\text{\textdegree}-60\text{\textdegree})=\\\\ =\sin 45\text{\textdegree}\cos 60\text{\textdegree}-\cos 45\text{\textdegree}\sin 60\text{\textdegree}=\\\\ =\dfrac{\sqrt 2}{2}\cdot \dfrac{1}{2}-\dfrac{\sqrt 2}{2}\cdot \dfrac{\sqrt 3}{2}=\\\\ =\dfrac{\sqrt 2}{4}-\dfrac{\sqrt 6}{4}=\\\\ =\dfrac{\sqrt 2-\sqrt 6}{4}

Sum and Difference Formulas for Cosine and Sine:

  1. cos(a±b)=cosacosbsinasinb\cos(a\pm b)=\cos a\cos b\mp \sin a\sin b

  2. sin(a±b)=sinacosb±cosasinb\sin (a\pm b)=\sin a\cos b\pm \cos a\sin b

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