Solve the following equations: 3 + =5 cosec x - Vidyadip

Question

Solve the following equations:
$3\tan\text{x}+\cot\text{x}=5\ \text{cosec }\text{x}$

✓

Answer

$3\tan\text{x}+\cot\text{x}=5\ \text{cosec}\ \text{x}$ $\Rightarrow\frac{3\sin\text{x}}{\cos\text{x}}+\frac{\cos\text{x}}{\sin\text{x}}=\frac{5}{\sin\text{x}}$ $\Rightarrow\frac{3\sin^2+\cos^2\text{x}}{\cos\text{x}\sin\text{x}}=\frac{5}{\sin\text{x}}$ $\Rightarrow3(1-\cos^2\text{x})+\cos^2\text{x}=5\cos\text{x}$ $\Rightarrow3-3\cos^2\text{x}+\cos^2\text{x}=5\cos\text{x}$ $\Rightarrow2\cos^2\text{x}+5\cos\text{x}-3=0$ $\Rightarrow2\cos^2\text{x}+6\cos\text{x}-\cos\text{x}-3=0$ $\Rightarrow2\cos\text{x}(\cos\text{x}+3)-1(\cos\text{x}+3)=0$ $\Rightarrow(2\cos\text{x}-1)(\cos\text{x}+3)=0$ $\Rightarrow(2\cos\text{x}-1)=0$ or $\cos\text{x}+3=0$ $\Rightarrow\cos\text{x}=\frac{1}{2}$ or $\cos\text{x}=-3$$\cos\text{x}=-3$ is not possible $(\therefore-1\leq\cos\text{x}\leq1)$
$\Rightarrow\cos\text{x}=\cos\frac{\pi}{3}$
$\Rightarrow\text{x}=2\text{n}\pi\pm\frac{\pi}{3},\ \text{n}\in\text{Z}$

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 "5 Marks Questions" from Trigonometric Equations→Trigonometric Equations — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Prove that the lines $\text{y}=\sqrt{3}\text{x}+1,\text{y}=4$ and $\text{y}=-\sqrt{3}\text{x}+2$ form an equilateral triangle.

$\frac{\sqrt{\sin\text{A}}-\sqrt{\sin\text{B}}}{\sqrt{\sin\text{A}}+\sqrt{\sin\text{B}}}=\frac{\text{a + b}-2\sqrt{\text{ab}}}{\text{a}-\text{b}}$

Prove the following by using the principle of mathematical induction for all n ∈ N:$\frac{1}{2.5}+\frac{1}{5.8}+\frac{1}{8.11}+...+\frac{1}{(3\text{n}-1)(3\text{n}+2)}=\frac{\text{n}}{(6\text{n}+4)}.$

Show that the straight lines $L_1 = (b + c)x + ay + 1 = 0, L_2 = (c + a)x + by + 1 = 0$ and $L_3 = (a + b)x + cy + 1 = 0$ are concurrent.

Prove that:
$\frac{\sin3\text{A}\cos4\text{A}-\sin\text{A}\cos2\text{A}}{\sin4\text{A}\sin\text{A}+\cos6\text{A}\cos\text{A}}=\tan2\text{A}$

Prove the following:
$\cos\Big(\frac{3\pi}{4}+\text{x}\Big)-\cos\Big(\frac{3\pi}{4}-\text{x}\Big)=-\sqrt{2}\sin\text{x}$

The sum of first two terms of an infinite G.P. is 5 and each term is three times the sum of the succeeding terms. Find the G.P.

The following are some particulars of the distribution of weights of boys and girls in a class:

Number	Boys	Girls
	100	50
Mean weight	60kg	45kg
Variance	9	4

Which of the distributions is more variable?

If $\text{T}_\text{n}=\sin^\text{n}\text{x}+\cos^\text{n}\text{x},$ Prove that
$6\text{T}_{10}-15\text{T}_8+10\text{T}_6-1=0$

The product of three numbers in G.P. is 216. If 2, 8, 6 be added to them, the results are in A.P. Find the numbers.