How do you express $cos (\frac{π}{3}) \cdot sin (\frac{11 π}{8})$ $cos(pi/ 3 ) * sin( ( 11 pi) / 8 )$ without using products of trigonometric functions?

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How do you express $cos (\frac{π}{3}) \cdot sin (\frac{11 π}{8})$ $cos(pi/ 3 ) * sin( ( 11 pi) / 8 )$ without using products of trigonometric functions?

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Answer 1 · 2016-04-15T21:11:05

$= \frac{1}{2} [sin (\frac{41 π}{24}) + sin (\frac{25 π}{24})]$ $=1/2 [sin ((41pi)/24) + sin ((25pi)/24)]$

Explanation:

Use formula

$sin (A + B) - sin (A - B) = 2 cos A sin B$ $sin (A+B) - sin (A-B) = 2 cos A sin B$

$cos A sin B = \frac{1}{2} [sin (A + B) - sin (A - B)]$ $cosAsinB=1/2 [sin (A+B) - sin (A-B)]$

$A = \frac{π}{3} and B = \frac{11 π}{8}$ $A=pi/3 and B=(11pi)/8$

$cos (\frac{π}{3}) sin (\frac{11 π}{8}) = \frac{1}{2} [sin (\frac{π}{3} + \frac{11 π}{8}) - sin (\frac{π}{3} - \frac{11 π}{8})]$ $cos(pi/3)sin((11pi)/8)=1/2 [sin (pi/3+(11pi)/8) - sin (pi/3-(11pi)/8)]$

$= \frac{1}{2} [sin (\frac{41 π}{24}) - sin (\frac{- 25 π}{24})]$ $=1/2 [sin ((41pi)/24) - sin ((-25pi)/24)]$

$= \frac{1}{2} [sin (\frac{41 π}{24}) + sin (\frac{25 π}{24})]$ $=1/2 [sin ((41pi)/24) + sin ((25pi)/24)]$

Answer 2 · 2016-04-16T03:11:35

$- (\frac{1}{2}) sin (\frac{3 π}{8})$ $- (1/2)sin ((3pi)/8)$

Explanation:

$P = cos (\frac{π}{3}) . sin (\frac{11 π}{8})$ $P = cos (pi/3).sin ((11pi)/8)$
Trig table --> $cos (\frac{π}{3}) = \frac{1}{2}$ $cos (pi/3) = 1/2$ .
$sin (\frac{11 π}{8}) = sin (\frac{3 π}{8} + π) = - sin (\frac{3 π}{8})$ $sin ((11pi)/8) = sin ((3pi)/8 + pi) = - sin ((3pi)/8)$
The product can be expressed as
$P = - (\frac{1}{2}) sin (\frac{3 π}{8})$ $P = -(1/2)sin((3pi)/8)$

We can evaluate P by applying the identity: $cos 2 a = 1 - 2 {sin}^{2} a$ $cos 2a = 1 - 2sin^2 a$
$cos (\frac{6 π}{8}) = cos (\frac{3 π}{4}) = - \frac{\sqrt{2}}{2} = 1 - 2 {sin}^{2} (\frac{3 π}{8})$ $cos ((6pi)/8) = cos ((3pi)/4) = - sqrt2/2 = 1 - 2sin^2 ((3pi)/8)$
$2 {sin}^{2} (\frac{3 π}{8}) = 1 + \frac{\sqrt{2}}{2} = \frac{2 + \sqrt{2}}{2}$ $2sin^2 ((3pi)/8) = 1 + sqrt2/2 = (2 + sqrt2)/2$
${sin}^{2} (\frac{3 π}{4}) = \frac{2 + \sqrt{2}}{4}$ $sin^2 ((3pi)/4) = (2 + sqrt2)/4$
$sin (\frac{3 π}{8}) = \frac{\sqrt{2 + \sqrt{2}}}{2}$ $sin ((3pi)/8) = sqrt(2 + sqrt2)/2$ --> $sin (\frac{3 π}{8})$ $sin ((3pi)/8)$ is positive.
Finally,
$P = - (\frac{1}{2}) . sin (\frac{3 π}{8}) = - (\frac{1}{4}) \sqrt{2 + \sqrt{2}}$ $P = - (1/2).sin ((3pi)/8) = -(1/4)sqrt(2 + sqrt2)$

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How do you express $cos (\frac{π}{3}) \cdot sin (\frac{11 π}{8})$ $cos(pi/ 3 ) * sin( ( 11 pi) / 8 )$ without using products of trigonometric functions?

2 Answers

Explanation:

Explanation:

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How do you express $cos (\frac{π}{3}) \cdot sin (\frac{11 π}{8})$ $cos(pi/ 3 ) * sin( ( 11 pi) / 8 )$ without using products of trigonometric functions?