Advertisements
Advertisements
Question
Write the value of sin \[\frac{\pi}{12}\] sin \[\frac{5\pi}{12}\].
Advertisements
Solution
sin \[\frac{\pi}{12}\] sin \[\frac{5\pi}{12}\]
\[= \frac{1}{2} \times 2\left( \sin\frac{\pi}{12} \right) \left( \sin\frac{5\pi}{12} \right)\]
\[ = \frac{1}{2}\left[ \cos\left( \frac{\pi}{12} - \frac{5\pi}{12} \right) - \cos\left( \frac{\pi}{12} + \frac{5\pi}{12} \right) \right] \left[ \because 2\sin A \sin B = \cos(A - B) - \cos(A + B) \right]\]
\[ = \frac{1}{2}\left[ \cos\left( - \frac{\pi}{3} \right) - \cos\frac{\pi}{2} \right]\]
\[ = \frac{1}{2}\left( \frac{1}{2} - 0 \right)\]
\[ = \frac{1}{4}\]
APPEARS IN
RELATED QUESTIONS
Prove that:
Prove that:
cos 10° cos 30° cos 50° cos 70° = \[\frac{3}{16}\]
Prove that:
tan 20° tan 40° tan 60° tan 80° = 3
Prove that tan 20° tan 30° tan 40° tan 80° = 1.
Prove that:
sin 10° sin 50° sin 60° sin 70° = \[\frac{\sqrt{3}}{16}\]
Prove that:
sin 20° sin 40° sin 60° sin 80° = \[\frac{3}{16}\]
If α + β = \[\frac{\pi}{2}\], show that the maximum value of cos α cos β is \[\frac{1}{2}\].
Express each of the following as the product of sines and cosines:
sin 12x + sin 4x
Express each of the following as the product of sines and cosines:
cos 12x - cos 4x
Prove that:
sin 50° + sin 10° = cos 20°
Prove that:
sin 105° + cos 105° = cos 45°
Prove that:
sin 50° − sin 70° + sin 10° = 0
Prove that:
cos 80° + cos 40° − cos 20° = 0
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
`sin A + sin 2A + sin 4A + sin 5A = 4 cos (A/2) cos((3A)/2)sin3A`
Prove that:
cos 20° cos 100° + cos 100° cos 140° − 140° cos 200° = −\[\frac{3}{4}\]
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Write the value of the expression \[\frac{1 - 4 \sin 10^\circ \sin 70^\circ}{2 \sin 10^\circ}\]
Write the value of \[\sin\frac{\pi}{15}\sin\frac{4\pi}{15}\sin\frac{3\pi}{10}\]
Express the following as the sum or difference of sine or cosine:
`cos (7"A")/3 sin (5"A")/3`
Express the following as the product of sine and cosine.
sin 6θ – sin 2θ
Express the following as the product of sine and cosine.
cos 2θ – cos θ
Prove that:
tan 20° tan 40° tan 80° = `sqrt3`.
Prove that:
(cos α – cos β)2 + (sin α – sin β)2 = 4 sin2 `((alpha - beta)/2)`
Prove that:
`(cos 2"A" - cos 3"A")/(sin "2A" + sin "3A") = tan "A"/2`
Evaluate:
sin 50° – sin 70° + sin 10°
Find the value of tan22°30′. `["Hint:" "Let" θ = 45°, "use" tan theta/2 = (sin theta/2)/(cos theta/2) = (2sin theta/2 cos theta/2)/(2cos^2 theta/2) = sintheta/(1 + costheta)]`
