How do you express $sin (\frac{π}{4}) \cdot sin (\frac{π}{3})$ $sin(pi/ 4 ) * sin( ( pi) / 3 )$ without using products of trigonometric functions?

Question

How do you express $sin (\frac{π}{4}) \cdot sin (\frac{π}{3})$ $sin(pi/ 4 ) * sin( ( pi) / 3 )$ without using products of trigonometric functions?

1 Answer

Impact of this question

1497 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-05-20T05:54:08

$sin (\frac{π}{4}) sin (\frac{π}{3}) = \frac{1}{2} [cos (\frac{π}{12}) + cos (\frac{5 π}{12})]$ $sin(pi/4)sin(pi/3)=1/2[cos(pi/12)+cos((5pi)/12)]$

Explanation:

We know that $cos (A + B) = cos A cos B - sin A sin B$ $cos(A+B)=cosAcosB-sinAsinB$ ..............(1)

and $cos (A - B) = cos A cos B + sin A sin B$ $cos(A-B)=cosAcosB+sinAsinB$ ..............(2)

Now subtracting (2) from (1)

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

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

Hence, $sin (\frac{π}{4}) sin (\frac{π}{3}) = \frac{1}{2} cos ((\frac{π}{4}) - (\frac{π}{3})) - \frac{1}{2} cos ((\frac{π}{4}) + (\frac{π}{3}))$ $sin(pi/4)sin(pi/3)=1/2cos((pi/4)-(pi/3))-1/2cos((pi/4)+(pi/3))$

= $\frac{1}{2} cos (- \frac{π}{12}) - \frac{1}{2} cos (\frac{7 π}{12})$ $1/2cos(-pi/12)-1/2cos((7pi)/12)$ ,

but $cos (- x) = cos x$ $cos(-x)=cosx$ and $cos (π - x) = - cos x$ $cos(pi-x)=-cosx$ ,

hence above is equal to

$\frac{1}{2} cos (\frac{π}{12}) - \frac{1}{2} cos (π - \frac{5 π}{12})$ $1/2cos(pi/12)-1/2cos(pi-(5pi)/12)$

= $\frac{1}{2} cos (\frac{π}{12}) + \frac{1}{2} cos (\frac{5 π}{12}) = \frac{1}{2} [cos (\frac{π}{12}) + cos (\frac{5 π}{12})]$ $1/2cos(pi/12)+1/2cos((5pi)/12)=1/2[cos(pi/12)+cos((5pi)/12)]$

Science

Math

Humanities

... and beyond

How do you express $sin (\frac{π}{4}) \cdot sin (\frac{π}{3})$ $sin(pi/ 4 ) * sin( ( pi) / 3 )$ without using products of trigonometric functions?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do you express sin(pi/ 4 ) * sin( ( pi) / 3 ) sin(π4)⋅sin(π3)sin(pi/ 4 ) * sin( ( pi) / 3 ) without using products of trigonometric functions?

1 Answer

Explanation:

Related questions

Impact of this question

How do you express $sin (\frac{π}{4}) \cdot sin (\frac{π}{3})$ $sin(pi/ 4 ) * sin( ( pi) / 3 )$ without using products of trigonometric functions?