Sine Rule

Q.

Evaluate$\sin 60°\cos 30°+\sin 30°\cos 60°$

Q.

Prove that:

$\frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}=2SecA$

Q.

If $\tan \left(\theta \right)=\frac{x\sin \left(\varphi \right)}{1-x\cos \left(\varphi \right)}$ and $\tan \left(\varphi \right)=\frac{y\sin \left(\theta \right)}{1-y\cos \left(\theta \right)}$, then $\frac{x}{y}=$

1. $\frac{\sin \left(\varphi \right)}{\sin \left(\theta \right)}$

2. $\frac{\sin \left(\theta \right)}{\sin \left(\varphi \right)}$

3. $\frac{\sin \left(\theta \right)}{1-\cos \left(\theta \right)}$

4. $\frac{\sin \left(\theta \right)}{1-\cos \left(\varphi \right)}$

Q.

The upper part of a tree broken over by the wind makes an angle of ${30}^{\circ }$ with the ground and the distance from the root to the point where the top of the tree touches the ground is 15 m. Using sine rule, find the height of the tree.

Q.

Prove the following: $\tan \theta +\tan (90°-\theta )=sec\theta ·sec(90°-\theta )$

Q. In $△ABC,\frac{sinB}{sin(A+B)}=$

1. $\frac{b}{c}$
2. $\frac{b}{a+b}$
3. $\frac{c}{b}$
4. None of these

Q.

If $\sin A+\sin 2A=x$ and $\cos A+\cos 2A=y$, then $\left(x^2+y^2\right)\left(x^2+\right.\left.y^2-3\right)=$

1. $2y$

2. $y$

3. $3y$

4. None of these

Q.

If the angles of a triangle are in the ratio 3 : 4 : 5, then the sides are in the ratio

1. 

2. 3 : 4 :5

3. 

4. 

Q.

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

1. abc

2. a+b+c

3. None of these

4. 3abc

Q.

$(\sin \theta ,\cos \theta )$and$(3,2)$ lies on the same side of the line $x+y=1$, then $\theta$ lies between

1. $\left(0,\frac{\pi }{2}\right)$

2. $\left(0,\pi \right)$

3. $\left(\frac{\pi }{4},\frac{\pi }{2}\right)$

4. $\left(0,\frac{\pi }{4}\right)$

Q. Let $PQRS$ be a quadrilateral in a plane, where $QR=1,\angle PQR=\angle QRS={70}^{\circ },\angle PQS={15}^{\circ }$ and $\angle PRS={40}^{\circ }$.
If $\angle RPS={\theta }^{\circ },PQ=\alpha$ and $PS=\beta$, then the interval(s) that contain(s) the value of $4\alpha \beta \sin {\theta }^{\circ }$ is/are

1. $\left(C\right)(\sqrt{2},3)$
2. $\left(B\right)(1,2)$
3. $\left(D\right)(2\sqrt{2},3\sqrt{2})$
4. $\left(A\right)(0,\sqrt{2})$

Q. $ABCD$ is a trapezium such that $AB$ and $CD$ are parallel and $BC\perp CD$. If $\angle ADB=\theta ,BC=p$ and $CD=q$, then $AB$ is equal to :

1. $\frac{(p^2+q^2)\sin \theta }{p\cos \theta +q\sin \theta }$
2. $\frac{p^2+q^2\cos \theta }{p\cos \theta +q\sin \theta }$
3. $\frac{p^2+q^2}{p\cos \theta +q\sin \theta }$
4. $\frac{(p^2+q^2)\sin \theta }{(p\cos \theta +q\sin \theta {)}^2}$

Q.

$\text{If in any}\Delta ABC,\angle C={105}^{\circ },\angle B={45}^{\circ },a=2,\text{then find b.}$

Q.

In $\Delta ABC,if\angle C={90}^{\circ },\angle A={30}^{\circ },c=20$, then the values of a and b are

1. $10,10$

2. $10,10\sqrt{3}$

3. $5,5\sqrt{3}$

4. $8,8\sqrt{3}$

Q.

If ${\sin }^{-1}(A+iB)=x+iy$, then $\frac{A}{B}$=

1. $\frac{\tan x}{\tan hy}$

2. $\frac{\tan hx}{\tan y}$

3. $\frac{\tan hx}{\tan hy}$

4. $\frac{\cos x}{\cos hy}$

Q. In a triangle $ABC,$ if $\frac{a^2+b^2}{a^2-b^2}\sin (A-B)=1$ and the triangle is not right angled, then $\cos (A-B)=$

1. $\sin (\frac{C}{2}-\frac{\pi }{4})$
2. $\cos (\frac{C}{2}-\frac{\pi }{4})$
3. $\tan (\frac{C}{2}-\frac{\pi }{4})$
4. $\cot (\frac{C}{2}-\frac{\pi }{4})$

Q.

$\frac{c-bcosA}{b-ccosA}=\frac{cosB}{cosC}$

Q.

If $\cos 2\alpha =\frac{(3\cos 2\beta -1)}{(3-\cos 2\beta )}$, then $\tan \alpha$is equal to

1. $\sqrt{2}\tan \beta$

2. $\tan \beta$

3. $\sin 2\beta$

4. $\sqrt{2}cot\beta$

Q. The perimeter of a triangle PQR is six times the arithmetic mean of the sines of its angles. If a = 1, then $\angle A$ = .

1. $60°$
2. $90°$
3. $30°$
4. $45°$

Q.

If in a $∆\mathrm{ABC}$, $4\mathrm{sinA}=4\mathrm{sinB}=3\mathrm{sinC}$, then $\mathrm{cosC}$ is equal to

1. $\frac{1}{3}$

2. $\frac{1}{9}$

3. $\frac{1}{27}$

4. $\frac{1}{8}$

Q.

$\text{In}\Delta ABC,\text{if a = 18, b = 24 and c = 30 and}\angle C={90}^{\circ },\text{find sin A, sin B and sin C.}$

Q.

If angle theta is equal to 45 degrees can we substitute the value and continue with our solution

Q. In any triangle ABC, prove the following:

$a^2\sin (B-C)=(b^2-c^2)\sin A$

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.

In a triangle $ABC$, if $\frac{\cos A}{a}=\frac{\tan C}{c}$, then $\sin (B+C)$ is equal to

1. $\cos B\sin A$

2. $\cos AcosC$

3. $\cos A\sin B$

4. $\sin B\sin C$

Q.

$a^{2}sin(B-C)=(b^2-c^2)sinA$

Q.

$△ABC$ is a right-angled triangle with sides $a,b,c$ and smallest angle $\theta$. If $\frac{1}{a},\frac{1}{b},\frac{1}{c}$ are also the sides of the right-angle triangle then find $\sin \theta$.

1. $\sqrt{\frac{3-\sqrt{5}}{2}}$

2. $\frac{3-\sqrt{5}}{2}$

3. $\sqrt{\frac{3+\sqrt{5}}{2}}$

4. $\frac{3+\sqrt{5}}{2}$

Q. y=tan theta, if %error is measurement of 1% then find the error measurement by at angle theta=/4

Q.

Q. In $\Delta ABC,\sin C+\cos C+\sin (2B+C)-\cos (2B+C)=2\sqrt{2}.$ Then the $\Delta ABC$ is

1. Acute angled isosceles triangle
2. Right angled isosceles triangle
3. Equilateral triangle
4. Obtuse angled isosceles triangle