exact value of an expression

[lewch45]

With these 2 problems I need to find the exact value of an expression without using a calculator. Can someone help me with this???? Any help would be appreciated.

(1) (sin 195 deg)(cos 15 deg)


(2) (sin 11 pi/12)(sin 7pi/12)

 

[royhaas]

Use appropriate identities and relations.
(1) 195=180+15; 15=30/2
(2) Use formulas for \(\displaystyle cos(x+y), cos(x-y), x=11\pi/12,y=7\pi/12\).

 

[soroban]

Hello, lewch45!

I have a back-door approach to these problems . . .

Find the exact value without a calculator.

\(\displaystyle (1)\; (\sin 195^o)(\cos 15^o)\)

Note that: \(\displaystyle \,\sin195^o\,=\,-\sin15^o\)

So we have: \(\displaystyle \,-\sin15^o\cdot\cos15^o\:=\:-\frac{1}{2}(2\cdot\sin15^o\cdot\cos15^o)\;=\;-\frac{1}{2}(\sin30^o)\:=\:-\frac{1}{2}\left(\frac{1}{2}\right)\;=\;-\frac{1}{4}\)

\(\displaystyle (2)\;\left(\sin\frac{11\pi}{12}\right)\left(\sin\frac{7\pi}{12}\right)\)

We have: \(\displaystyle \,(\sin165^o)(\sin105^o)\)

Note that: \(\displaystyle \,\sin165^o\,=\,\sin15^o,\;\) and \(\displaystyle \,\sin105^o\,=\,\sin75^o\,=\,\cos15^o\)

So we have: \(\displaystyle \,\sin15^o\cdot\cos15^o\:=\:\frac{1}{2}\cdot\sin30^o\:=\:\frac{1}{2}\left(\frac{1}{2}\right)\:=\:\frac{1}{4}\)

 

[tkhunny]

With these 2 problems I need to find the exact value of an expression without using a calculator. Can someone help me with this???? Any help would be appreciated.

(1) (sin 195 deg)(cos 15 deg)


(2) (sin 11 pi/12)(sin 7pi/12)

I'm always a little humored by this type of problem.

(sin 195º)(cos 15º) <== That IS an exact value. Why does someone think it isn't?