Question
Solve the following equations: $\sin\theta+\sin5\theta=\sin3\theta$
✓
Answer
$\sin\text{x}+\sin5\text{x}=\sin3\text{x}$
$\Rightarrow2\sin3\text{x},\cos2\text{x}-\sin3\text{x}=0$
$\Big[\because\sin\text{C}+\sin\text{D}=2\sin\frac{\text{C}+\text{D}}{2}.\cos\frac{\text{C}-\text{D}}{2}\Big] $
Either
$\Rightarrow\sin3\text{x}=0$ or $2\cos2\text{x}-1=0 $
$\Rightarrow3\text{x}=\text{n}\pi,\text{n }\in\ \text{z}$ or $\cos2\text{x}=\frac{1}{2}=\cos\frac{\pi}{3}$
$\Rightarrow\text{x}=\frac{\text{n}\pi}{3},\text{n }\in\ \text{z}$ or $2\text{x}=2\text{m}\pi\pm\frac{\pi}{3},\text{m }\in\ \text{z}$
or $\text{x}=\text{m}\pi\pm\frac{\pi}{6}$
Thus,
$\text{x}=\frac{\text{n}\pi}{3}$ or $\text{m}\pi\pm\frac{\pi}{6},\text{n, m }\in\ \text{z}$
$\Rightarrow2\sin3\text{x},\cos2\text{x}-\sin3\text{x}=0$
$\Big[\because\sin\text{C}+\sin\text{D}=2\sin\frac{\text{C}+\text{D}}{2}.\cos\frac{\text{C}-\text{D}}{2}\Big] $
Either
$\Rightarrow\sin3\text{x}=0$ or $2\cos2\text{x}-1=0 $
$\Rightarrow3\text{x}=\text{n}\pi,\text{n }\in\ \text{z}$ or $\cos2\text{x}=\frac{1}{2}=\cos\frac{\pi}{3}$
$\Rightarrow\text{x}=\frac{\text{n}\pi}{3},\text{n }\in\ \text{z}$ or $2\text{x}=2\text{m}\pi\pm\frac{\pi}{3},\text{m }\in\ \text{z}$
or $\text{x}=\text{m}\pi\pm\frac{\pi}{6}$
Thus,
$\text{x}=\frac{\text{n}\pi}{3}$ or $\text{m}\pi\pm\frac{\pi}{6},\text{n, m }\in\ \text{z}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
The focus of a parabolic mirror as shown in is at a distance of 5 cm from its vertex. If the mirror is 45 cm deep, find the distance AB
Find the values of the other five trigonometric functions in the following:
$\sin\text{x}=\frac{3}{5},$ x in quadrant I
→$\sin\text{x}=\frac{3}{5},$ x in quadrant I
The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of 2nd hour, 4th hour and nth hour?
Using binomial theorem, write down the expansions of the following:
$\Big(\text{x}-\frac{1}{\text{x}}\Big)^6$
A rail road curve is to be laid out on a circle. What radius should be used if the track is to change direction by 25° in a distance of 40 metres?
Let A = {1, 2, 3} and B = {3, 4}. Find A × B and show it graphically.
Prove that: $\cos 2 x \cos \frac{x}{2}-\cos 3 x \cos \frac{9 x}{2}$ $=\sin 5 x \sin \frac{5 x}{2}$
→In a group of 1000 people, there are 750 who can speak Hindi and 400 who can speak Bengali. How many can speak Hindi only? How many can speak Bengali? How many can speak both Hindi and Bengali?
Find the two numbers whose A.M. is 25 and G.M. is 20.
Find the equation to the ellipse whose centre is (-2, 3) and semi-axis are 3 and 2 when major axis is
Parallel to y-axis.