Solve the following equations: 3 2x-5 x + 8 2x = 2 - Vidyadip

Question

Solve the following equations: $3\sin2\text{x}-5 \sin\text{x}\cos \text{x} + 8 \cos2\text{x = 2}$

✓

Answer

$3\sin^2\text{x}-5\sin\text{x}\cos\text{x}+8\cos^2\text{x}=2$ $\Rightarrow3\sin^2\text{x}-5\sin\text{x}\cos\text{x}+3\cos^\text{x}2+5\cos^2\text{x}-2=0$ $\Rightarrow3(\sin^2\text{x}\cos^2\text{x})-5\sin\text{x}\cos\text{x}+5\cos^2\text{x}-2=0$ $\Rightarrow3-5\sin\text{x}\cos\text{x}+5\cos^2\text{x}-2=0$ $\Rightarrow5\cos^2\text{x}-5\sin\text{x}\cos\text{x}+1=0$ $\Rightarrow5(1-\sin^2\text{x})-5\sin\text{x}\cos\text{x}+1=0$ $\Rightarrow5-5\sin^2\text{x}-5\sin\text{x}\cos\text{x}+1=0$ $\Rightarrow5\sin^2\text{x}+5\sin\text{x}\cos\text{x}-6=0$ Dividing by $\cos^2\text{x},\ $ we get $\Rightarrow5\tan^2\text{x}+5\tan\text{x}-6\sec^2\text{x}=0$ $\Rightarrow5\tan^2\text{x}+5\tan\text{x}-6-6\tan^2\text{x}=0$ $\Rightarrow-\tan^2\text{x}+5\tan\text{x}-6=0$ $\Rightarrow\tan^2\text{x}-5\tan\text{x}+6=0$ $\Rightarrow\tan^2\text{x}-3\tan\text{x}-2\tan\text{x}+6=0$ $\Rightarrow(\tan\text{x}-3)=0$ or $\tan\text{x}=2$ $\Rightarrow\text{x}=\text{n}\pi+\tan^{-1}3$ or $\text{x}=\text{n}\pi+\tan6{-1}2,\ \text{n}\in\text{Z}$

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 mean deviation from the median for the following data:

Mark obtained	10	11	12	14	15
No. of students	2	3	8	3	4

Find the sum of the following series to n terms: $1 × 2 + 2 × 3 + 3 × 4 + 4 × 5 + ...$

Calculate the mean deviation about the median of the following observation: $38, 70, 48, 34, 63, 42, 55, 44, 53, 47$

Solve the following systems of linear inequations graphically: $\text{x}-\text{y}\leq1,\text{x}+2\text{y}\leq8,2\text{x}+\text{y}\geq2,\text{x}\geq0$ and $\text{y}\geq0$

Prove the following: $\cos^22\text{x}-\cos^26\text{x}=\sin8\text{x}\sin4\text{x}$

A point R with x-coordinate 4 lies on the line segment joining the points P(2, –3, 4) and Q(8, 0, 10). Find the coordinates of the point R. [Hint Suppose R divides PQ in the ratio k : 1. The coordinates of the point R are given by $\Big(\frac{8\text{k}+2}{\text{k}+1},\frac{-3}{\text{k}+1},\frac{10\text{k}+4}{\text{k}+1}\Big)]$

Reduce the equation 3x - 2y + 6 = 0 to the intercept form and find the x and y intercepts.

If $\cos\text{x}=-\frac{3}{5}$ and x lies in the IIIrd quadrant, find the values of $\cos\frac{\text{x}}{2},\sin\frac{\text{x}}{2},\sin2\text{x}.$

Prove that: $\frac{\sin(\theta+\phi)-2\sin\theta+\sin(\theta-\phi)}{\cos(\theta+\phi)-2\cos\theta+\cos(\theta-\phi)}=\tan\theta$

If A = {1, 2, 3}, B = {4}, C = {5}, then verify that:
$\text{A}\times(\text{B}\cup\text{C})=(\text{A}\times\text{B})\cup(\text{A}\times\text{C})$