Solution of Triangle

Q.

What is the value of $\cos 15°$?

Q.

What is the value of $\cos 75°$?

Q.

Find the value of $\tan {45}^0$.

Q.

$\frac{(\cos 9°+\sin 9°)}{(\cos 9°-\sin 9°)}$ is equal to

1. $\tan 26°$

2. $\tan 81°$

3. $\tan 51°$

4. $\tan 54°$

5. $\tan 46°$

Q.

In a triangle $ABC,$the value of $\sin \left(A\right)+\sin \left(B\right)+\sin \left(C\right)$ is :

1. $4\sin \left(\frac{A}{2}\right)\sin \left(\frac{B}{2}\right)\sin \left(\frac{C}{2}\right)$

2. $4\cos \left(\frac{A}{2}\right)\cos \left(\frac{B}{2}\right)\cos \left(\frac{C}{2}\right)$

3. $4\cos \left(\frac{A}{2}\right)\sin \left(\frac{B}{2}\right)\sin \left(\frac{C}{2}\right)$

4. $4\cos \left(\frac{A}{2}\right)\sin \left(\frac{B}{2}\right)\cos \left(\frac{C}{2}\right)$

Q.

$\frac{1}{4}[\surd 3\cos 23°-\sin 23°]=$

1. $\cos 43°$

2. $\cos 7°$

3. $\cos 53°$

4. $Noneofthese$

Q.

$\left(\frac{1}{\cos 80°}-\frac{\sqrt{3}}{\sin 80°}\right)$ is equal to

1. $\sqrt{2}$

2. $\sqrt{3}$

3. $2$

4. $4$

5. $\sqrt{5}$

Q.

If $A=130°$ and $x=\sin A+\cos A$, then,

1. $x>0$

2. $x<0$

3. $x=0$

4. $x\le 0$

Q.

The value of $\cot 105°$ is

1. $\sqrt{3}-2$

2. $2-\sqrt{3}$

3. $\sqrt{2}+3$

4. $\sqrt{3}+2$

Q.

The value of $4(\cos ²60°+\cos ²45°)-\sin ²30°$ is

Q. If $a,b,c$ are the sides of the $\Delta ABC$ and $a^2,b^2,c^2$ are the roots of $x^3-p{x}^2+qx-k=0,$ then

1. $\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}=\frac{P}{2\sqrt{k}}$
2. $a\cos A+b\cos B+c\cos C=\frac{4q-p^2}{2\sqrt{k}}$
3. $a\sin A+b\sin B+c\sin C=\frac{2p\Delta }{\sqrt{k}}$
4. $\sin A\sin B\sin C=\frac{8{\Delta }^3}{k}$

Q. Match the following by appropriately matching the lists based on the information given in $\text{Column I}$ and $\text{Column II}$.

$\begin{array}{cc}ColumnI& ColumnII\left(Typeof△ABC\right)\\ \text{a.}\cot \frac{A}{2}=\frac{b+c}{a}& \text{p. always right angled}\\ \text{b.}a\tan A+b\tan B=(a+b)\tan \frac{A+B}{2}& \text{q. always isosceles}\\ \text{c.}a\cos A=b\cos B& \text{r. may be right angled}\\ \text{d.}\cos A=\frac{\sin B}{2\sin C}& \text{s. may be right angled isosceles}\end{array}$

1. $a-p,q;b-q,r;c-p,s;d-q,r$
2. $a-p,q;b-q,s;c-p,s;d-q,s$
3. $a-p,s;b-q,r;c-p,s;d-q,r$
4. $a-p,s;b-q,r,s;c-r,s;d-q,r,s$

Q.

If $co{s}^{2}A+co{s}^{2}C=si{n}^{2}B,$ then $△$ ABC is
[MP PET 1991]

1. Equilateral

2. Right angled

3. Isosceles

4. None of these

Q. Number of triangles possible for a given b, c and B(acute angle) under the condition that b < c sin B. Where b, c are the sides and B is the angle opposite to b.

1. \N
2. 1
3. 2
4. 3

Q. In a triangle $PQR$, let $\angle PQR={30}^{\circ }$ and the sides $PQ$ and $QR$ have lengths $10\sqrt{3}$ and $10$, respectively. Then, which of the following statement(s) is (are) TRUE?

1. $\angle QPR={45}^{\circ }$
2. The area of the triangle $PQR$ is $25\sqrt{3}$ and $\angle QRP={120}^{\circ }$
3. The radius of the incircle of the triangle $PQR$ is $10\sqrt{3}-15$
4. The area of the circumcircle of the triangle $PQR$ is $100\pi$

Q. If b > c sin B, b < c and B is acute angle then number of triangles possible following the given conditions is 1.

1. True
2. False

Q. In a triangle $ABC,a:b:c=4:5:6$. The ratio of the radius of the circumcircle to that to the incircle is

1. $15/4$
2. $11/5$
3. $16/7$
4. $16/3$

Q.

$\frac{\cos 30°+\sin 60°}{1+\cos 60°+\sin 30°}$ is equals to:

1. $\frac{1}{2}$

2. $\frac{1}{\sqrt{2}}$

3. $1$

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

Q. In $\Delta$ ABC having vertices $A(acos{\theta }_1,asin{\theta }_1),B(acos{\theta }_2,asin{\theta }_2)$ and $C(acos{\theta }_3,asin{\theta }_3)$ is equilateral, then which of the followings is/are true?

1. $cos{\theta }_1+cos{\theta }_2+cos{\theta }_3=0$
2. $sin{\theta }_1+sin{\theta }_2+sin{\theta }_3=0$
3. $cos({\theta }_1-{\theta }_2)+cos({\theta }_2-{\theta }_3)+cos({\theta }_3-{\theta }_1)=\frac{-3}{2}$
4. $\left|sin\left(\frac{{\theta }_1-{\theta }_2}{2}\right)\right|=sin\left|\left(\frac{{\theta }_2-{\theta }_3}{2}\right)\right|=\left|sin\left(\frac{{\theta }_1-{\theta }_3}{2}\right)\right|$

Q.

If the median of $\Delta ABC$ through $A$ is perpendicular to $AB$ then

1. $tanA+tanB=0$

2. $2tanA+tanB=0$

3. $tanA+2tanB=0$

4. None of these

Q. In a right angled triangle, altitude divides the hypotenuse into two segments of length $8$ units and $18$ units, then the length of the smallest side is
(correct answer + 1, wrong answer - 0.25)

1. $8\sqrt{2}\text{units}$
2. $6\sqrt{13}\text{units}$
3. $4\sqrt{13}\text{units}$
4. $8\sqrt{3}\text{units}$