Prove that: 20^ 40^ 80^ = 3 8 - Vidyadip

Question

Prove that:
$\sin20^\circ\sin40^\circ\sin80^\circ=\frac{\sqrt3}{8}$

✓

Answer

$\text{LHS}=\sin20^\circ\sin40^\circ\sin80^\circ$
$=\ \frac{1}{2}(2\sin20^\circ\sin40^\circ)\sin80^\circ$
$=\ \frac{1}{2}[\cos(40^\circ-20^\circ)-\cos(40^\circ+20^\circ)]\sin80^\circ$$[\because2\sin\text{A}\sin\text{B}=\cos(\text{A}-\text{B})-\cos(\text{A+B})$
$=\ \frac{1}{2}[\cos20^\circ-\cos60^\circ]\sin80^\circ$
$=\ \frac{1}{2}\Big[\cos20^\circ-\frac{1}{2}\Big]\sin80^\circ$
$=\ \frac{1}{2}[\cos20^\circ\sin80^\circ]-\frac{1}{4}\sin80^\circ$
$=\ \frac{1}{4}[2\cos20^\circ\sin80^\circ-\sin80^\circ]$
$=\ \frac{1}{4}[\sin(80^\circ+20^\circ)+\sin(80^\circ-20^\circ)-\sin80^\circ]$$[\because2\sin\text{A}\cos\text{B}=\sin(\text{A+B})+\sin(\text{A}-\text{B})$
$=\ \frac{1}{4}[\sin100^\circ+\sin60^\circ-\sin80^\circ]$
$=\ \frac{1}{4}\Big[\sin(180^\circ-80^\circ)+\frac{\sqrt3}{2}-\sin80^\circ\Big]$
$=\ \frac{1}{4}\Big[\sin80^\circ+\frac{\sqrt3}{2}-\sin80^\circ\Big]$
$= \frac{\sqrt3}{8}=\ \text{RHS}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Transformation Formulae→Transformation Formulae — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Differentiate the following from first principle$\sqrt{\tan\text{x}}$

Prove that:
$\text{If}\cos\text{A}+\cos\text{B}=\frac{1}{2}\text{ and }\sin\text{A}+\sin\text{B}=\frac{1}{4},$ prove that $\tan\Big(\frac{\text{A+B}}{2}\Big)=\frac{1}{2}.$

Find the equation to the ellipse whose foci are $(-4, 0)$, and $(-4, 0),$ eccentricity = $\frac{1}{3}.$

Find the distance of the point of intersection of the lines $2x + 3y = 21$ and $3x - 4y + 11 = 0$ from the line $8x + 6y + 5 = 0$.

Differentiate the following from the first principle$\text{e}^{\sqrt{\text{ax}+\text{b}}}$

Sum the following series to n terms:
$3 + 5 + 9 + 15 + 23 + .....$

Prove the following by the principle of mathematical induction:
n(n + 1)(n + 5) is a multiple of 3 for all $\text{n}\in\text{N}$

Find the angles between the following pairs of straight lines:
3x + y + 12 = 0 and x + 2y - 1 = 0

Find the coefficient of variation for the following data:

Size (in cms):	10-15	15-20	20-25	25-30	30-35	35-40
No. of items:	2	8	20	35	20	15

Planes are drawn through the points (5, 0, 2) and (3, -2, 5) parallel to the coordinate planes. Find the lengths of the edges of the rectangular parallelopiped so formed.