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slamgirl [31]
2 years ago
9

Find the exact value by using a half-angle identity. sine of five pi divided by twelve

Mathematics
2 answers:
xeze [42]2 years ago
7 0

Answer : The exact value of \sin (\frac{5\pi}{12})=\frac{\sqrt{(2+\sqrt{3})}}{2}

Step-by-step explanation :

As we are given that: \sin (\frac{5\pi}{12})

Using a half-angle identity:

\sin\left(\frac{\theta}{2}\right)=\sqrt{\frac{1-\cos\theta}{2}}

\sin\left(\frac{\theta}{2}\right)=\sin \frac{(\frac{5\pi}{6})}{2}

Thus, \theta=\frac{5\pi}{6}

Now using half-angle identity, we get:

\sin\left(\frac{\theta}{2}\right)=\sqrt{\frac{1-\cos\theta}{2}}

\sin\left(\frac{(\frac{5\pi}{6})}{2}\right)=\sqrt{\frac{1-\cos (\frac{5\pi}{6})}{2}}

\sin\left(\frac{(\frac{5\pi}{6})}{2}\right)=\sqrt{\frac{1-\cos (\pi -\frac{\pi}{6})}{2}}

\sin\left(\frac{(\frac{5\pi}{6})}{2}\right)=\sqrt{\frac{1+\cos (\frac{\pi}{6})}{2}}

As we know that,

\cos \frac{\pi}{6}=\cos 30^o=\frac{\sqrt{3}}{2}

Now put the value of \cos \frac{\pi}{6}, we get:

\sin (\frac{5\pi}{12})=\sqrt{\frac{1+(\frac{\sqrt{3}}{2})}{2}}

\sin (\frac{5\pi}{12})=\sqrt{\frac{(\frac{2+\sqrt{3}}{2})}{2}}

\sin (\frac{5\pi}{12})=\sqrt{(\frac{2+\sqrt{3}}{4})}

\sin (\frac{5\pi}{12})=\frac{\sqrt{(2+\sqrt{3})}}{2}

Thus, the exact value of \sin (\frac{5\pi}{12})=\frac{\sqrt{(2+\sqrt{3})}}{2}

maks197457 [2]2 years ago
3 0

Answer:

cos

(

5

π

12

)

=

√

2

−

√

3

2

Explanation:

By the half angle formula:

XXXX

cos

(

θ

2

)

=

±

√

1

+

cos

(

θ

)

2

If  

θ

2

=

5

π

12

XXXX

then  

θ

=

5

π

6

Note that  

5

π

6

is a standard angle in quadrant 2 with a reference angle of  

π

6

so  

cos

(

5

π

6

)

=

−

cos

(

π

6

)

=

−

√

3

2

Therefore

XXXX

cos

(

5

π

12

)

=

±







⎷

1

−

√

3

2

2

XXXXXXXXXXX

=

±







⎷

2

−

√

3

2

2

XXXXXXXXXXX

=

±

√

2

−

√

3

4

XXXXXXXXXXX

=

±

√

2

−

√

3

2

Since  

5

π

12

<

π

2

XXXX

5

π

12

is in quadrant 1

XXXX

→

cos

(

5

π

12

)

is positive

XXXX

XXXX

XXXX

(the negative solution is extraneous)

answer in pic more explain

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