Solve the following equations: + 2x+ 3x+ 4x=0 - Vidyadip

Question

Solve the following equations: $\sin\text{x}+\sin2\text{x}+\sin3\text{x}+\sin4\text{x}=0$

✓

Answer

Given, $\sin\text{x}+\sin2\text{x}+\sin3\text{x}+\sin4\text{x}=0$$(\sin4\text{x}+\sin2\text{x})+(\sin3\text{x}+\sin\text{x})=0$
Using, $(\sin\text{A}+\sin\text{B})\text{Formula}\Rightarrow$
$2\sin\Big[\frac{(4\text{x}+2\text{x})}{2}\Big]\cos\Big[\frac{4\text{x}-2\text{x}}{2}\Big]+2\sin\Big[\frac{( 3\text{x}+\text{x})}{2}\Big]\cos\Big[\frac{( 3\text{x}-\text{x})}{2}\Big]=0$
$2\sin3\text{x}\cos\text{x}+2\sin2\text{x}\cos\text{x}=0$
$2\cos\text{x}(\sin3\text{x}+\sin2\text{x})=0$
$2\cos\text{x}(2\sin)\Big[\frac{(3\text{x}+2\text{x})}{2}\Big]\cos\Big[\frac{(3\text{x}-2\text{x})}{2}\Big]=0$
$4\cos\text{x}\sin\frac{5\text{x}}{2}\cos\frac{\text{x}}{2}=0$
$\cos\text{x}=0;\sin\frac{5\text{x}}{2}=0;\cos\frac{\text{x}}{2}=0$
$\text{x}=\frac{(2\text{n}+1)\pi}{2};\frac{5\text{x}}{2}=\text{m}\pi;\frac{\text{x}}{2}=\frac{(2\text{x}+1)\pi}{2}$
$\text{x}=\frac{(2\text{n}+1)\pi}{2};\text{x}=\frac{2\text{m}\pi}{5};\text{x}=(2\text{x}+1)\pi,\text{m},\text{r},\text{n}\in\text{Z}$

Need a full question paper?

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

Start Generating Free

Explore more

More "(Each question 4 marks)" from Trigonometric Equations→Trigonometric Equations — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the angle in radians through which a pendulum swings if its length is 75cm and the tip describes an arc of length.

10cm
15cm
21cm

Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0}\frac{3\sin^2\text{x}-2\sin\text{x}^2}{3\text{x}^2}$

Find the standard deviation for the following data:

x	2	3	4	5	6	7
f	4	9	16	14	11	6

Find the image of the point (2, 1) with respect to the line mirror x + y - 5 = 0.

Prove that: $\tan82\frac{1^\circ}{2}=(\sqrt{3}+\sqrt{2})(\sqrt{2}+1)=\sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{6}$

Prove that the straight lines (a + b)x + (a - b )y = 2ab, (a - b)x + (a + b)y = 2ab and x + y = 0 form an isosceles triangle whose vertical angle is $2\tan^{-1}\Big(\frac{\text{a}}{\text{b}}\Big).$

The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is $87\frac{1}{2}.$ Find them.

Find the equations of the lines joining the vertex of the parabola $y^2 = 6x$ to the point on it which have abscissa $24$.

For any two sets of A and B, prove that:

$\text{B}'\subset\text{A}'\Rightarrow\text{A}\subset\text{B.}$

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): $\frac{\sin\text{x}+\cos\text{x}}{\sin\text{x}-\cos\text{x}}$