Question
Evaluate the following:
$\frac{\sin30^\circ-\sin90^\circ+2\cos0^\circ}{\tan30^\circ\tan60^\circ}$
$\frac{\sin30^\circ-\sin90^\circ+2\cos0^\circ}{\tan30^\circ\tan60^\circ}$
✓
Answer
We have,
$\frac{\sin30^\circ-\sin90^\circ+2\cos0^\circ}{\tan30^\circ\tan60^\circ}\ \dots(1)$
Now,
$\sin30^\circ=\frac{1}{2},\ \sin90^\circ=\cos0^\circ=1,$ $\tan30^\circ=\frac{1}{\sqrt{3}},\tan60^\circ=\sqrt{3}$
So by substituting above values in equation (1)
We get,
$\frac{\sin30^\circ-\sin90^\circ+2\cos0^\circ}{\tan30^\circ\tan60^\circ}$
$=\frac{\frac{1}{2}-1+2\times1}{\frac{1}{\sqrt{3}}\times\sqrt{3}}$
Now, $\sqrt{3}$ present in the denominator of above expression gets cancelled and we get,
$\frac{\sin30^\circ-\sin90^\circ+2\cos0^\circ}{\tan30^\circ\tan60^\circ}$
$=\frac{\frac{1}{2}-1+2}{1}$
$=\frac{1}{2}-1+2$
Now by taking LCM in the above expression we get,
$\frac{\sin30^\circ-\sin90^\circ+2\cos0^\circ}{\tan30^\circ\tan60^\circ}$
$=\frac{1}{2}-\frac{1\times2}{1\times2}+\frac{2\times2}{1\times2}$
$=\frac{1}{2}-\frac{2}{2}+\frac{4}{2}$
$=\frac{1-2+4}{2}$
$=\frac{5-2}{2}$
$=\frac{3}{2}$
Therefore,
$\frac{\sin30^\circ-\sin90^\circ+2\cos0^\circ}{\tan30^\circ\tan60^\circ}=\frac{3}{2}$
$\frac{\sin30^\circ-\sin90^\circ+2\cos0^\circ}{\tan30^\circ\tan60^\circ}\ \dots(1)$
Now,
$\sin30^\circ=\frac{1}{2},\ \sin90^\circ=\cos0^\circ=1,$ $\tan30^\circ=\frac{1}{\sqrt{3}},\tan60^\circ=\sqrt{3}$
So by substituting above values in equation (1)
We get,
$\frac{\sin30^\circ-\sin90^\circ+2\cos0^\circ}{\tan30^\circ\tan60^\circ}$
$=\frac{\frac{1}{2}-1+2\times1}{\frac{1}{\sqrt{3}}\times\sqrt{3}}$
Now, $\sqrt{3}$ present in the denominator of above expression gets cancelled and we get,
$\frac{\sin30^\circ-\sin90^\circ+2\cos0^\circ}{\tan30^\circ\tan60^\circ}$
$=\frac{\frac{1}{2}-1+2}{1}$
$=\frac{1}{2}-1+2$
Now by taking LCM in the above expression we get,
$\frac{\sin30^\circ-\sin90^\circ+2\cos0^\circ}{\tan30^\circ\tan60^\circ}$
$=\frac{1}{2}-\frac{1\times2}{1\times2}+\frac{2\times2}{1\times2}$
$=\frac{1}{2}-\frac{2}{2}+\frac{4}{2}$
$=\frac{1-2+4}{2}$
$=\frac{5-2}{2}$
$=\frac{3}{2}$
Therefore,
$\frac{\sin30^\circ-\sin90^\circ+2\cos0^\circ}{\tan30^\circ\tan60^\circ}=\frac{3}{2}$
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
|
Class interval
|
50-70
|
70-90
|
90-110
|
110-130
|
130-150
|
150-180
|
|
Frequency
|
18
|
12
|
13
|
27
|
8
|
22
|
Given the linear equation $2x + 3y - 8 = 0$, write another linear equation in two variables such that the geometrical representation of the pair so formed is:
→- Intersecting lines.
- Parallel lines.
- Coincident lines.
Solve the following systems of equations:$\frac{1}{3\text{x}+\text{y}}+\frac{1}{3\text{x}-\text{y}}=\frac{3}{4}$
$\frac{1}{2(3\text{x}-\text{y})}+\frac{1}{2(3\text{x}-\text{y})}=\frac{-1}{8}$
→$\frac{1}{2(3\text{x}-\text{y})}+\frac{1}{2(3\text{x}-\text{y})}=\frac{-1}{8}$
Three coins are tossed together. Find the probability of getting:
- Exactly two heads.
- At most two heads.
- At least one head and one tail.
- No tails.
Area of a sector of central angle $200^\circ $ of a circle s $770cm^2$. Find the length of the corresponding are of this sector.
→A train travels at a certain average speed for a distance of $54\ km$ and then travels a distance of $63\ km$ at an average of $6\ km/hr$ more than the first speed. If it takes $3$ hours to complete the total journey, what is its first speed?
→Fill in the blanks given in the contract note of sale-purchase of shares.
(B - buy S - sell)
(B - buy S - sell)
Prove the following trigonometric identities.
$\frac{1-\cos\text{A}}{1+\cos\text{A}}=(\cot\text{A}-\text{cosec A})^2$
→$\frac{1-\cos\text{A}}{1+\cos\text{A}}=(\cot\text{A}-\text{cosec A})^2$
On an environment day, students in a school planted 120 trees under plantation project. The information regarding the project is shown in the following table. Show it by a pie diagram.
| Tree name | Karanj | Behada | Arjun | Bakul | Kadunimb |
| No. of trees | 20 | 28 | 24 | 22 | 26 |
If A(5,4 ) , B (-3 , -2) and C (1 -8) are the vertices of a ∆ ??? . Segment AD is median. Find the length of seg AD: