Suppose \(\theta = \frac{11\pi}{12}\). How do you use the sum identity to find the exact value of \(\sin \theta\)?
Answer (2)
The better way is, first we have to find the equivalent in degrees
2 Ο = 360 \ΒΊ
12 11 Ο β = 345 \ΒΊ
now we can change this value to β 15 \ΒΊ
how do we get an angle like this?!
30 \ΒΊ β 45 \ΒΊ = β 15 \ΒΊ
then
s in ( 30 \ΒΊ β 45 \ΒΊ ) = s in ( 30 \ΒΊ ) β cos ( 45 \ΒΊ ) β s in ( 45 \ΒΊ ) β cos ( 30 \ΒΊ )
\begin{Bmatrix}sin(30\ΒΊ)&=&\frac{1}{2}\\\\sin(45\ΒΊ)&=&cos(45\ΒΊ)&=&\frac{\sqrt{2}}{2}}\end{matrix}\\\\cos(30\ΒΊ)&=&\frac{\sqrt{3}}{2}\end{matrix}
now we replace this values
s in ( β 15 \ΒΊ ) = 2 1 β β 2 2 β β β 2 2 β β β 2 3 β β
s in ( β 15 \ΒΊ ) = 4 2 β β β 4 6 β β
s in ( β 15 \ΒΊ ) = s in ( 345 \ΒΊ ) = 4 2 β β 6 β β β β
To find sin ( 12 11 Ο β ) , we use the sine sum identity. By expressing 12 11 Ο β as 3 Ο β + 4 Ο β , we can find that sin ( 12 11 Ο β ) = 4 6 β + 2 β β .
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