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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

$- \frac{1}{16}$

$\cos (\frac{π}{15}) \cos (\frac{2 π}{15}) \cos (\frac{4 π}{15}) \cos (\frac{8 π}{15}) = \frac{1}{4} (2 \cos (\frac{4 π}{15}) \cos (\frac{π}{15})) (2 \cos (\frac{8 π}{15}) \cos (\frac{2 π}{15})) = \frac{1}{4} (\cos (60 °) + \cos (36 °)) (\cos (120 °) + \cos (72 °)) = \frac{1}{4} (\frac{1}{2} + \frac{\sqrt{5} + 1}{4}) (- \frac{1}{2} + \frac{\sqrt{5} - 1}{4}) [∵ \cos (36 °) = \frac{\sqrt{5} + 1}{4} and \cos (72 °) = \frac{\sqrt{5} - 1}{4}] = \frac{1}{4} [- \frac{1}{4} + \frac{1}{2} (\frac{\sqrt{5} - 1}{4} - \frac{\sqrt{5} + 1}{4}) + \frac{5 - 1}{16}] = \frac{1}{4} [\frac{1}{2} (- \frac{1}{2})] = - \frac{1}{16}$