Sine Rule

Q. mod sin x+mod cos x=mod sin x+cos x

Q.

If the angles of a triangle be in the ratio 1 : 2 : 7, then the ratio of its greatest side to the least side is

1. 1:2

2. 2:1

3. $(\sqrt{5}+1):(\sqrt{5}-1)$

4. $(\sqrt{5}-1):(\sqrt{5}+1)$

Q. If the angles of a triangle are in the ratio 1:2:3, then the ratio of its corresponding sides is .

1. $\sqrt{3}:2:2\sqrt{3}$
2. $1:2:2\sqrt{3}$
3. $1:\sqrt{3}:2$
4. $1:\sqrt{3}:2\sqrt{3}$

Q.

If $a=2\sqrt{2},b=6,A={45}^0,$then

1. two triangle are possible

2. either no triangle or two triangles are possible

3. no triangle is possible

4. one triangle is possible

Q.

If in a $△ABC,si{n}^{3}A+si{n}^{3}B+si{n}^{3}C=3sinAsinBsinC$ then the value of the determinant
$\left[\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right]$ is

1. None of these

2. 

3. (a+b+c)(ab+bc+ca)

4. 0

Q. If in a triangle $ABC,a,b,c$ are in A.P. and $p_1,p_2,p_3$ are the altitudes from the vertices $A,B,C$ respectively then

1. $p_1,p_2,p_3$ are in A.P.
2. $p_1,p_2,p_3$ are in H.P.
3. $p_1+p_2+p_3\le \frac{3R}{\Delta }$
4. $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\le \frac{3R}{\Delta }$

Q.

In a$\Delta$ABC, if $(\sqrt{3}-1)a=2b,A=3B,$ then $C$ is

1. ${30}^{\circ }$

2. ${45}^{\circ }$

3. ${60}^{\circ }$

4. ${120}^{\circ }$

Q. The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one, then the sides are

1. $6,7,8$
2. $4,5,6$
3. $1,2,3$
4. $3,4,5$

Q.

In a triangle $ABC\angle A={60}^{\circ },\angle B={40}^{\circ }$ and $\angle C={80}^{\circ }$ If P is the centre of the circumcircle of triangle $ABC$ with radius unity, then the radius of the circumcircle of triangle $BPC$ is

1. 1

2. $\sqrt{3}$

3. 2

4. $\frac{\sqrt{3}}{2}$

Q.

In a triangle $ABC,a^{3}cos(B-C)+b^{3}cos(C-A)+c^{3}cos(A-B)=$

1. abc

2. 3abc

3. a+b+c

4. None of these

Q. In ∆ABC, prove that:

$\frac{b\sec B+c\sec C}{\tan B+\tan C}=\frac{c\sec C+a\sec A}{\tan C+\tan A}=\frac{a\sec A+b\sec B}{\tan A+\tan B}$

Q. In a $\Delta ABC$; the value of $\frac{co{s}^2\frac{B-C}{2}}{(b+c{)}^2}+\frac{si{n}^2\frac{B-C}{2}}{(b-c{)}^2}$ is

1. $a^2$
2. $\frac{1}{a^2}$
3. $\frac{b^2+c^2}{a^2}$
4. $\frac{b^2-c^2}{a^2}$

Q.

In a $\Delta$ ABC, a, c, A are given and $b_1,b_2$ are two values of the third side b such that $b_2,2b_1$ , then sin A is equal to

1. $\sqrt{\frac{9a^2-c^2}{8c^2}}$

2. $\sqrt{\frac{9a^2-c^2}{8c^2}}$

3. $\sqrt{\frac{9a^2+c^2}{8c^2}}$

4. None of these

Q.

(i) In any $\Delta ABC$, prove that $\frac{sin(B-C)}{sin(B+C)}=\frac{b^2-c^2}{a^2}$

(ii) Prove that $cosAcos({60}^{\circ }-A)cos({60}^{\circ }+A)=\frac{1}{4}cos3A$

Or

Find the value of

$\left(i\right)tan\frac{13\pi }{12}$

$\left(ii\right)sin{360}^{\circ }$

$\left(iii\right)cos{18}^{\circ }$

Q. In $△ABC$ with usual notations, $\frac{a\cos A+b\cos B+c\cos C}{a+b+c}=\frac{1}{2},$ then the value of $\frac{\sin A+\sin B+\sin C}{\sin A\sin B\sin C}$ is

Q. Write the value of $\cos \frac{\pi }{7}\cos \frac{2\pi }{7}\cos \frac{4\pi }{7}.$

Q. In a triangle ABC, if $\frac{\cos A}{a}=\frac{\cos B}{b}=\frac{\cos C}{c}$, then triangle ABC will be:

1. Isosceles triangle
2. Equilateral triangle
3. Right triangle
4. Right Isosceles triangle

Q.

The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Then the sides of the triangle are

1. 1, 2, 3

2. 2, 3, 4

3. 3, 4, 5

4. 4, 5, 6

Q. In ∆ABC, if a², b² and c² are in A.P., prove that cot A, cot B and cot C are also in A.P.

Q.

In $\Delta ABC,ccos(A-\alpha )+\alpha cos(C+\alpha )=$

1. $\alpha cos\alpha$

2. $bcos\alpha$

3. $ccos\alpha$

4. $2bcos\alpha$

Q. A triangle $ABC$ is such that sides $a,b,c$ are in $G.P.$ and $\sin A,\sin B,\sin C$ are in $A.P.,$ then the $△ABC$ is
(where $a,b,c$ are sides (in $\text{units}$) of $△ABC$ opposite to $\angle A,\angle B$ and $\angle C$ respectively)

1. scalene triangle
2. None of these
3. Equilateral triangle
4. Right angled triangle

Q.

If two sides of a triangle are “a” units & 10 units and the angles opposite of them are 30°and 60° respectively then find the value of a .

1. 5
2. 10
3. $\frac{10}{\sqrt{3}}$
4. $10\sqrt{3}$

Q. If tan A + tan B = a and cot A + cot B = b, prove that cot (A + B) $\frac{1}{a}-\frac{1}{b}$.

Q.

in triangle abc if cot a + cot b + cot c = root 3 then triangle abc is