Question

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

Answer 1

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

Step-by-step explanation :

As we are given that:
`\sin ((5\pi)/(12))`

Using a half-angle identity:

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

`\sin\left((\theta)/(2)\right)=\sin (((5\pi)/(6)))/(2)`

Thus,
`\theta=(5\pi)/(6)`

Now using half-angle identity, we get:

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

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

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

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

As we know that,

`\cos (\pi)/(6)=\cos 30^o=(√(3))/(2)`

Now put the value of
`\cos (\pi)/(6)`, we get:

`\sin ((5\pi)/(12))=\sqrt{(1+((√(3))/(2)))/(2)}`

`\sin ((5\pi)/(12))=\sqrt{(((2+√(3))/(2)))/(2)}`

`\sin ((5\pi)/(12))=\sqrt{((2+√(3))/(4))}`

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

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

Answer 2

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