Trigonometric Ratios for Sum of Two Angles

Q.

Find the value of tan A + 2 tan2A + 4 tan4A + 8 cot 8A

1. tanA

2. cotA

3. secA

4. cosecA

Q. If $\alpha$ and $\beta$ are distinct roots of $acos\theta +bsin\theta =c$, Prove that $sin(\alpha +\beta )=\frac{2ab}{a^2+b^2}$.
or
Prove that $cos{20}^{\circ }cos{40}^{\circ }cos{60}^{\circ }cos{80}^{\circ }=\frac{1}{16}$

Q.

If cos 3theta = $\alpha cos\theta +\beta co{s}^3\theta$, then $(\alpha ,\beta )$ =

1. (3, 4)

2. (4, 3)

3. (-3, 4)

4. (3, -4)

Q. If A lies in the third quadrant and 3 tan A - 4 = 0, then 5 sin 2A + 3 sin A + 4 cos A =

1. 
2. 
3. 
4. 

Q.

The point A(sin $\theta$, cos$\theta$) is 3 units away from the point B (2 cos ${75}^{\circ }$, 2 sin ${75}^{\circ }$). If $0^{\circ }$ $\le \theta <$ ${360}^{\circ }$, then $\theta$ is __ degree

Q.

If $cos(\theta -\alpha )$ = a, $sin(\theta -\beta )$ = b,

then $co{s}^2(\alpha -\beta )$ + 2ab $sin(\alpha -\beta )$ is equal to

1. 

2. 

3. 

4. 

Q.

Convert the complex number z=-1-i in to the polar form.

Q.

In an equlilateral triangle, (circumradius) : (inradius) : (exradius) is equal to

1. 1 : 1 : 1

2. 1 : 2 : 3

3. 2 : 1 : 3

4. 3 : 2 : 4

Q.

1 - 2$si{n}^2(\frac{\pi }{4}+\theta )$ =

1. cos2θ

2. -cos2θ

3. sin2θ

4. -sin2θ

Q.

If ${\tan }^{-1}x+{\cos }^{-1}\frac{y}{\sqrt{1+y^2}}={\sin }^{-1}\frac{3}{\sqrt{10}}$ and both $x$ and $y$ are positive and integral, then $x$ and $y$ are equal to

1. $(1,2)$ and $(2,7)$

2. $(1,2)$ and $(1,7)$

3. $(1,7)$ and $(2,7)$

4. $(1,2)$ and $(2,1)$

Q.

If in a triangle ABC, a = 15, b = 36, c = 39, then sin C/2 is equal to

1. √3/2

2. 1/2

3. 1/√2

4. -1/√2

Q.

$\frac{1}{tan3A-tanA}$ - $\frac{1}{cot3A-cotA}$ =

1. tanA

2. cot2A

3. cotA

4. tan2A

Q.

$\frac{d}{dx}\left[log|secx+tanx|\right]=$

1. cot x

2. sec x

3. 

4. tan x

Q.

In a triangle ABC, tan A + tan B +tan C = 6 and tan A $\times$ tan B = 2, then the value of tan A, tan B and tan C are

1. 1, 2, 3

2. 2, 1, 3

3. 1, 2, 0

4. 1, 2, 2

Q. $\sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{6}$ is equal to

1. $cot7\frac{1^{\circ }}{2}$
2. $sin7\frac{1^{\circ }}{2}$
3. $sin{15}^{\circ }$
4. $cos{15}^{\circ }$

Q. The value of ${\tan }^2{27}^{\circ }+2\tan {27}^{\circ }\tan {36}^{\circ }$ is equal to

Q. The value of ${\sum }_{k=1}^{13}\frac{1}{sin(\frac{\pi }{4}+\frac{(k-1)\pi )}{6})sin(\frac{\pi }{4}+\frac{k\pi }{6})}$ is equal to

1. $2(3-\sqrt{3})$
2. $2(\sqrt{3}-1)$
3. $2(2-\sqrt{3})$
4. $3-\sqrt{3}$

Q. If 3 $\cot \theta$ = 2, find the value of $\frac{4\sin \theta -3\cos \theta }{2\sin \theta +6\cos \theta }$

Q.

If $cos2B=\frac{cos(A+C)}{cos(A-C)}$ then

1. tanA, tanB, tanC are in AP

2. tanA, tanB, tanC are in HP

3. tanA, tanB, tanC are in AGP

4. tanA, tanB, tanC are in GP

Q.

The minimum value of $cos\theta +sin\theta$ is

[MNR 1976; Pb. CET 1996]

1. 0

2. 

3. 1/2

4. 

Q.

Which of the following are correct?

(1) cot(A+B) = $\frac{cotAcotB-1}{cotA+cotB}$

(2) cot(A-B) = $\frac{cotAcotB-1}{cotA-cotB}$

(3) tan(A+B) = $\frac{tanA+tanB}{1-tanAtanB}$

(4) tan(A-B) = $\frac{tanA-tanB}{1-tanAtanB}$

1. None of them

2. Only (1) and (2)

3. Only (1) and (3)

4. All the four

Q.

If in a triangle ABC, a = 15, b = 36, c = 39, then sin C/2 is equal to

1. √3/2

2. 1/√2

3. -1/√2

4. 1/2

Q. Prove that $(\sec A+\tan A-1)(\sec A-\tan A+1)-2\tan A=0$

Q. The value of ${\sum }_{k=1}^{13}\frac{1}{sin(\frac{\pi }{4}+\frac{(k-1)\pi )}{6})sin(\frac{\pi }{4}+\frac{k\pi }{6})}$ is equal to

1. $3-\sqrt{3}$
2. $2(3-\sqrt{3})$
3. $2(\sqrt{3}-1)$
4. $2(2-\sqrt{3})$

Q.

$\frac{1}{tan3A-tanA}-\frac{1}{cot3A-cotA}=$

1. tan2A

2. cot2A

3. 

4. 

Q. If $\tan A,\tan B,\tan C$ are the roots of the equation $2x^3-5x^2-2x+4=0,$ then the value of $\tan (A+B+C)$ is

1. $\frac{7}{4}$
2. $\frac{1}{4}$
3. $\frac{9}{4}$
4. undefined

Q.

$x=n\pi +\frac{3\pi }{4}$ or $x=m\pi +ta{n}^{-1}\frac{1}{2}$, where m, $n\in I$

Q.

If (a-b) sin($\theta +\varphi$) = (a+b) sin($\theta -\varphi$), find tan$\theta$

1. 

2. 

3. 

4. 

Q.

For $\frac{-\pi }{2}$ < $\theta$ < $\frac{\pi }{2}$ , $\frac{sin\theta +sin2\theta }{1+cos\theta +cos2\theta }$ lies in the interval.

1. (, )

2. (-2, 2)

3. (0, )

4. (-1, 1)

Q. If $tan\theta =\frac{4}{5}$ then find the value of $\frac{5sin\theta -3cos\theta }{5sin\theta +3cos\theta }$.