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:
cos 40° cos 80° cos 160° = \[- \frac{1}{8}\]
Prove that:
cos 20° cos 40° cos 80° = \[\frac{1}{8}\]
Prove that:
sin 10° sin 50° sin 60° sin 70° = \[\frac{\sqrt{3}}{16}\]
Prove that:
\[\tan x \tan \left( \frac{\pi}{3} - x \right) \tan \left( \frac{\pi}{3} + x \right) = \tan 3x\]
Express each of the following as the product of sines and cosines:
sin 2x + cos 4x
Prove that:
cos 100° + cos 20° = cos 40°
Prove that:
sin 40° + sin 20° = cos 10°
Prove that:
sin 50° − sin 70° + sin 10° = 0
Prove that:
Prove that:
Prove that:
cos A + cos 3A + cos 5A + cos 7A = 4 cos A cos 2A cos 4A
Prove that:
cos 20° cos 100° + cos 100° cos 140° − 140° cos 200° = −\[\frac{3}{4}\]
Prove that:
Prove that:
Prove that:
If cosec A + sec A = cosec B + sec B, prove that tan A tan B = \[\cot\frac{A + B}{2}\].
Prove that:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0
If (cos α + cos β)2 + (sin α + sin β)2 = \[\lambda \cos^2 \left( \frac{\alpha - \beta}{2} \right)\], write the value of λ.
If cos A = m cos B, then write the value of \[\cot\frac{A + B}{2} \cot\frac{A - B}{2}\].
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), then write the value of tan A tan B tan C.
The value of cos 52° + cos 68° + cos 172° is
If sin α + sin β = a and cos α − cos β = b, then tan \[\frac{\alpha - \beta}{2}\]=
The value of sin 50° − sin 70° + sin 10° is equal to
If cos A = m cos B, then \[\cot\frac{A + B}{2} \cot\frac{B - A}{2}\]=
If \[\tan\alpha = \frac{x}{x + 1}\] and
Express the following as the product of sine and cosine.
cos 2θ – cos θ
Prove that:
`(cos 2"A" - cos 3"A")/(sin "2A" + sin "3A") = tan "A"/2`
Prove that:
`(cos 7"A" +cos 5"A")/(sin 7"A" −sin 5"A")` = cot A
Prove that cos 20° cos 40° cos 60° cos 80° = `3/16`.
Evaluate:
sin 50° – sin 70° + sin 10°
If cos A + cos B = `1/2` and sin A + sin B = `1/4`, prove that tan `(("A + B")/2) = 1/2`
If cosec A + sec A = cosec B + sec B prove that cot`(("A + B"))/2` = tan A tan B.
