Trigonometric Ratios of Compound Angles

Q.

What is the exact value of $\cos 135°$ $?$

Q.

If $\cos (\alpha +\beta )=\frac{4}{5}$, $\sin (\alpha -\beta )=\frac{5}{13}$ and $\alpha ,\beta$ lie between $0$ and $\frac{\pi }{4}$, then $\tan 2\alpha =?$

1. $\frac{16}{63}$

2. $\frac{56}{33}$

3. $\frac{28}{33}$

4. None of these

Q.

Find the value of $\sin {135}^{\circ }$.

Q.

How do I find the value of $\sin \left(225°\right)$.

Q. $\frac{cos{10}^{\circ }+sin{10}^{\circ }}{cos{10}^{\circ }-sin{10}^{\circ }}$ is equal to

1. $-cot{35}^{\circ }$
   
2. $tan{55}^{\circ }$
   
3. $cot{55}^{\circ }$
   
4. $-tan{35}^{\circ }$
   

Q. If $\sin \theta +{\sin }^2\theta =1,$ then ${\cos }^2\theta +{\cos }^4\theta =$

1. $\sqrt{2}$
2. 1
3. 2
4. 0

Q.

Prove that $\frac{\mathrm{sin\theta }}{1-\mathrm{cos\theta }}+\frac{\mathrm{tan\theta }}{1+\mathrm{cos\theta }}=\mathrm{sec\theta }.\mathrm{cosec\theta }+\mathrm{cot\theta }$

Q. 11. sin(A+B) = sinA.cosB + cosA.sinB prove by vector method.

Q. Let $S=\frac{1}{\sin 8^{\circ }}+\frac{1}{\sin {16}^{\circ }}+\frac{1}{\sin {32}^{\circ }}+\dots \dots +\frac{1}{\sin {4096}^{\circ }}+\frac{1}{\sin {8192}^{\circ }}$. If $S=\frac{1}{\sin \alpha }$, where $\alpha \in (0,{90}^{\circ })$, then $\alpha$ (in degree) is

Q.

What is the difference between $2\cos x$ and $\cos 2x$?

Q.

If $x$ takes non-positive permissible value, then ${\sin }^{-1}x=$

1. ${\cos }^{-1}\sqrt{1-x^2}$

2. $-{\cos }^{-1}\sqrt{1-x^2}$

3. ${\cos }^{-1}\sqrt{x^2-1}$

4. $\mathrm{\pi }-{\cos }^{-1}\sqrt{1-x^2}$

Q.

If $sinA=\frac{1}{2},$ $cosB=\frac{12}{13},$ where $\frac{\pi }{2}<A<\pi$ and $\frac{3\pi }{2}<B<2\pi ,$ find tan(A-B)

Q.

The formula of $\cot (\mathrm{A}-\mathrm{B})$ is?

Q.

assuming that a person of a normal sight can read print at such a distance that the letters subtend an angle of 5' at his eye, find the height of letter that he can read a distance of 12m

Q. The value of ${\sin }^{2}52{\frac{1}{2}}^{\circ }-{\sin }^{2}22{\frac{1}{2}}^{\circ }$ is

1. $\frac{\sqrt{3}-1}{2\sqrt{2}}$
2. $\frac{\sqrt{3}+1}{2\sqrt{2}}$
3. $\frac{\sqrt{3}-1}{4\sqrt{2}}$
4. $\frac{\sqrt{3}+1}{4\sqrt{2}}$

Q.

Prove that $\frac{(1-\sin A)}{(1+\sin A)}={(secA-\tan A)}^2$

Q. The domain of the function $\sqrt{\sin 2x}$ is

1. $\cup [n\pi -\frac{\pi }{2},n\pi ],n\in Z$
2. $\cup [n\pi ,n\pi +\frac{\pi }{2}],n\in Z$
3. $\cup \left[\right(2n-1)\pi ,2n\pi ],n\in Z$
4. $\cup [2n\pi ,(2n+1\left)\pi \right],n\in Z$

Q.

The value of $\frac{tanx}{tan3x}$ whenever defined never lie between

1. 

2. 

3. 

4. 

Q. If $cosP=\frac{1}{7}$ and $cosQ=\frac{13}{14},$ where P and Q both are acute angles. Then the values of P-Q is

1. ${30}^{\circ }$
   
2. ${60}^{\circ }$
   
3. ${45}^{\circ }$
   
4. ${75}^{\circ }$

Q.

Prove that, $\frac{1+\cos A}{\sin A}=\frac{\sin A}{1-\cos A}$

Q. If $\tan {40}^{\circ }+2\tan {10}^{\circ }=\cot x$, where $x\in (0,\pi /2)$, then the possible value of $x$ is

1. ${75}^{\circ }$
2. ${85}^{\circ }$
3. ${30}^{\circ }$
4. ${40}^{\circ }$

Q.

Prove that:

$si{n}^2(\frac{\pi }{8}+\frac{A}{2})-si{n}^2(\frac{\pi }{8}-\frac{A}{2})=\frac{1}{\sqrt{2}}sinA$

Q.

Prove that: $cos{105}^{\circ }+cos{15}^{\circ }=sin{75}^{\circ }-sin{15}^{\circ }$

Q.

If $cot\left({\cos }^{-1}x\right)=sec\left[{\tan }^{-1}\left(\frac{a}{\sqrt{b^2-a^2}}\right)\right]$, then $x=$

1. $\frac{b}{\sqrt{b^2–a^2}}$

2. $\frac{a}{\sqrt{2b^2–a^2}}$

3. $\frac{\sqrt{2b^2–a^2}}{a}$

4. $\frac{\sqrt{2b^2–a^2}}{b}$

Q. If $f\left(A\right)=\sum _{r=1}^8\tan rA\cdot \tan (r+1)A$ then

1. $f\left({18}^{\circ }\right)=-10$
2. $f\left({36}^{\circ }\right)=-10$
3. $f\left(\frac{\pi }{4}\right)=-8$
4. $f\left(\frac{\pi }{8}\right)=-8$

Q.

if cot 20=p then [tan 160-tan110/]/1+tan 160*tan 110

Q. If $\cos A=\frac{5}{13},\tan B=-\frac{15}{8}$, where $A$ is in $IV$ quadrant and $B$ is in $II$ quadrant, then $A+B$ belongs to which quadrant ?

1. $IV$
2. $III$
3. $II$
4. $I$

Q.

Find the integrals of the functions.
$\int$ sin x sin 2x sin 3x dx

Q. The value of $2({\cos }^{2}73°+{\cos }^{2}47°)-\cos 154°$ is

1. $0$
2. $-1$
3. $1$
4. $2$

Q.

A vertical pole consists of two parts, the lower part being one-third of the whole. At a point in the horizontal plane through the base of the pole and distance $20$ meters from it, the upper part of the pole subtends an angle whose tangent is $\frac{1}{2}$. The possible heights of the pole are

1. $20m$ and $20\sqrt{3}m$

2. $20m$ and $60m$

3. $16m$ and $48m$

4. none of these