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Trigonometric Functions

Multiple Choice Questions

701.

The smallest value of $5 \cos (θ) + 12$ is

Short Answer Type

702.

Show that

$\frac{\sin (θ)}{\cos (3 θ)} + \frac{\sin (3 θ)}{\cos (9 θ)} + \frac{\sin (9 θ)}{\cos (27 θ)} = \frac{1}{2} ((\tan 27 θ) - \tan (θ))$

Multiple Choice Questions

703.

The equation $\sqrt{3} \sin (x) + \cos (x) = 4$ has

infinitely many solutions
no solution
two solutions
only one solution

704.

The value of

$\tan (α) + 2 \tan (2 α) + 4 \tan (4 α) + . . . + 2^{n - 1} \tan (2^{n - 1} α) + 2^{n} \cot (2^{n} α)$ is

$\cot (2^{n} α)$
$2^{n} \tan (2^{n} α)$
0
$\cot (α)$

705.

If $\tan (\frac{απ}{4}) = \cot (\frac{βπ}{4})$ , then

$α + β = 0$
$α + β = 2 n$
$α + β = 2 n + 1$
$α + β = 2 (2 n + 1), \forall n is an integer$

706.

The principal value of $\sin^{- 1} (\tan (- \frac{5 π}{4}))$ is

$\frac{π}{4}$
- $\frac{π}{4}$
$\frac{π}{2}$
$- \frac{π}{2}$

707.

The value of $\cos (\frac{π}{15}) \cos (\frac{2 π}{15}) \cos (\frac{4 π}{15}) \cos (\frac{8 π}{15})$

$\frac{1}{16}$
$- \frac{1}{16}$
1
0

708.

The principal amplitude of ${(\sin (40 °) + icos (40 °))}^{5}$

70 $°$
- 110 $°$
110°
- 70°

Short Answer Type

709.

Find the general solution of $\sec (θ) + 1 = (2 + \sqrt{3}) \tan (θ)$

Multiple Choice Questions

710.
The real part of ${(1 - \cos (θ) + 2 isin (θ))}^{- 1}$

$\frac{1}{3 + 5 \cos (θ)}$

$\frac{1}{5 - 3 \cos (θ)}$

$\frac{1}{3 - 5 \cos (θ)}$

$\frac{1}{5 + 3 \cos (θ)}$

$\frac{1}{5 + 3 \cos (θ)}$

Given, ${\{(1 - \cos (θ)) + 2 isin (θ)\}}^{- 1}$

$= {(2 \sin^{2} (\frac{θ}{2}) + i . 4 \sin (\frac{θ}{2}) \cos (\frac{θ}{2}))}^{- 1} = {(2 \sin (\frac{θ}{2}))}^{- 1} {(\sin (\frac{θ}{2}) + i 2 \cos (\frac{θ}{2}))}^{- 1} = {(2 \sin (\frac{θ}{2}))}^{- 1} \frac{1}{\sin (\frac{θ}{2}) + i 2 \cos (\frac{θ}{2})} \times \frac{\sin (\frac{θ}{2}) - i 2 \cos (\frac{θ}{2})}{\sin (\frac{θ}{2}) - i 2 \cos (\frac{θ}{2})} = {(2 \sin (\frac{θ}{2}))}^{- 1} \frac{\sin (\frac{θ}{2}) - i 2 \cos (\frac{θ}{2})}{(\sin^{2} (\frac{θ}{2}) + 4 \cos^{2} (\frac{θ}{2}))}$

$= \frac{\sin (\frac{θ}{2}) - i 2 \cos (\frac{θ}{2})}{2 \sin (\frac{θ}{2}) (\sin^{2} (\frac{θ}{2}) + 4 \cos^{2} (\frac{θ}{2}))} = \frac{\sin (\frac{θ}{2}) - i 2 \cos (\frac{θ}{2})}{2 \sin (\frac{θ}{2}) (1 + 3 \cos^{2} (\frac{θ}{2}))}$

$It' s real part = \frac{\sin (\frac{θ}{2})}{2 \sin (\frac{θ}{2}) (1 + 3 \cos^{2} (\frac{θ}{2}))} = \frac{1}{2 \{1 + 3 (\frac{\cos (θ) + 1}{2})\}} = \frac{1}{2 + 3 \cos (θ) + 3} = \frac{1}{5 + 3 \cos (θ)}$