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Trigonometric Functions

Multiple Choice Questions

911.
If sin(A) + sin(B) + sin(C) = 0 and cos(A) + cos(B) + cos(C) = 0, then cos (A + B) + cos (B + C) + cos(C + A) is equal to

cos(A + B + C)

2

1

0

$Let z_{1} = \cos (A) + isin (A) z_{2} = \cos (B) + isin (B) and z_{3} = \cos (C) + isin (C) Then, z_{1} = \cos (A) - isin (A) z_{2} = \cos (B) - isin (B) and z_{3} = \cos (C) - isin (C) Now, if z = \cos (θ) + isin (θ) z = \frac{(\cos (θ) - isin (θ)) (\cos (θ) + isin (θ))}{(\cos (θ) + isin (θ))} = \frac{\cos^{2} (θ) + \sin^{2} (θ)}{\cos (θ) + isin (θ)} = \frac{1}{\cos (θ) + isin (θ)} = \frac{1}{z}$

$Now, z_{1} + z_{2} + z_{3} = (\cos (A) - isin (A)) + (\cos (B) - isin (B)) + (\cos (C) - isin (C)) = (\cos (A) + \cos (B) + \cos (C)) - i (\sin (A) + \sin (B) + \sin (C)) \Rightarrow \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} = 0 [∵ z = \frac{1}{z}] \Rightarrow z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1} = 0 \Rightarrow \sum (\cos (A) + isin (A)) (\cos (B) + isin (B)) = 0 \Rightarrow \sum (\cos (A) \cos (B) - \sin (A) \sin (B)) + (\sin (A) \cos (B) + \cos (A) \sin (B)) = 0 \Rightarrow \sum \cos (A + B) + isin (A + B) = 0 \Rightarrow \sum \cos (A + B) = 0 [∵ comparing the real part] \Rightarrow \cos (A + B) + \cos (B + C) + \cos (C + A) = 0$

912.

If $\tan (θ) . \tan (120 ° - θ) \tan (120 ° + θ) = \frac{1}{\sqrt{3}}$ , then $θ$ is equal to

$\frac{nπ}{3} + \frac{π}{18}, n \in Z$
$\frac{nπ}{3} + \frac{nπ}{12}, n \in Z$
$\frac{nπ}{12} + \frac{π}{12}, n \in Z$
$\frac{nπ}{3} + \frac{π}{6}, n \in Z$

913.

${(\frac{1 + \cos (\frac{π}{8}) - isin (\frac{π}{8})}{1 + \cos (\frac{π}{8}) + isin (\frac{π}{8})})}^{8} = ?$

1
- 1
2
$\frac{1}{2}$

914.

$If (1 + \tan (α)) (1 + \tan (4 α)) = 2, α \in (0, \frac{π}{16}), then α = ?$

$\frac{π}{20}$
$\frac{π}{30}$
$\frac{π}{40}$
$\frac{π}{50}$

915.

$If \cos (θ) = \frac{\cos (α) - \cos (β)}{1 - \cos (α) \cos (β)}, then one of the values of \tan (\frac{θ}{2}) is$

$\cot (\frac{β}{2}) \tan (\frac{α}{2})$
$\tan (\frac{α}{2}) \tan (\frac{β}{2})$
$\tan (\frac{β}{2}) \cot (\frac{α}{2})$
$\tan^{2} (\frac{α}{2}) \tan^{2} (\frac{β}{2})$

916.

The value of the expression $\frac{1 + \sin (2 α)}{\cos (2 α - 2 π) \tan (α - \frac{3 π}{4})} - \frac{1}{4} \sin (2 α) [\cot (\frac{α}{2}) + \cot (\frac{3 π}{2} + \frac{α}{2})] is$

0
1
$\sin^{2} (\frac{α}{2})$
$\sin^{2} (α)$

917.

$If \frac{1}{6} \sin (θ), \cos (θ) and \tan (θ) are in geometric progression, then the solution set of θ is$

$2 nπ \pm (\frac{π}{6})$
$2 nπ \pm (\frac{π}{3})$
$nπ + {(- 1)}^{n} (\frac{π}{3})$
$nπ + (\frac{π}{3})$

918.

$In ∆ ABC if x = \tan (\frac{B - C}{2}) \tan (\frac{A}{2}), y = \tan (\frac{C - A}{2}) \tan (\frac{B}{2}) and z = \tan (\frac{A - B}{2}) \tan (\frac{C}{2}), then (x + y + z) = ?$

xyz
- xyz
2xyz
$\frac{1}{2} xyz$

919.

$If A > 0, B > 0 and A + B = \frac{π}{3}, then the maximum value of Atan (B) is$

$\frac{1}{\sqrt{3}}$
$\frac{1}{3}$
$\frac{1}{2}$
$\sqrt{3}$

920.

$In ∆ ABC, if bcos (θ) = c - a, (where θ is an acute angle), then (c - a) \tan (θ) = ?$

$2 \sqrt{ca} \cos (\frac{B}{2})$
$2 \sqrt{ac} \sin (\frac{B}{2})$
$2 cacos (\frac{B}{2})$
$2casin (\frac{B}{2})$

Previous Year Papers

Test Series

Test Yourself

Multiple Choice Questions

911. If sin(A) + sin(B) + sin(C) = 0 and cos(A) + cos(B) + cos(C) = 0, then cos (A + B) + cos (B + C) + cos(C + A) is equal to cos(A + B + C) 2 1 0

911.
If sin(A) + sin(B) + sin(C) = 0 and cos(A) + cos(B) + cos(C) = 0, then cos (A + B) + cos (B + C) + cos(C + A) is equal to

cos(A + B + C)

2

1

0