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Trigonometric Functions

Multiple Choice Questions

821.

$The expression for \tan (9 °) - \tan (27 °) - \tan (63 °) + \tan (81 °) is equal to$

822.

$In ∆ ABC, \cos (\frac{B + 2 C + 3 A}{2}) + \cos (\frac{A - B}{2}) is$

823.

$The value of series \cos (12 °) + \cos (84 °) + \cos (132 °) + \cos (156 °) is$

$\frac{1}{2}$
$\frac{1}{4}$
$- \frac{1}{4}$
$- \frac{1}{2}$

824.

$For x \in IR, 3 \cos (4 x - 5) + 4 lies in the interval$

[1, 7]
[4, 7]
[0, 7]
[2, 7]

825.

If $x = \log [\cot (\frac{π}{4} + θ)]$ , then the value of $\sin (hx)$ is

$\tan (2 θ)$
$- \tan (2 θ)$
$\cot (2 θ)$
$- \cot (2 θ)$

826.

If in a $∆$ ABC, r₃ = r₁ + r₂ + r, then $∠ A + ∠ B$ is equal to

120°
100°
90°
80°

827.

In a $∆ ABC$ ${(a - b)}^{2} \cos^{2} (\frac{C}{2}) + {(a + b)}^{2} \sin^{2} (\frac{C}{2})$ is equal to

a²
c²
b²
a² + b²

828.
In a $∆ ABC$ , the correct formulae among the following are
$I . r = 4 Rsin (\frac{A}{2}) \sin (\frac{B}{2}) \sin (\frac{C}{2}) II . r_{1} = (s - a) \tan (\frac{A}{2}) III . r_{3} = \frac{∆}{s - c}$

only I, II

only II, III

only I, III

I, II, III

only I, III

$I . Since, 4 Rsin (\frac{A}{2}) \sin (\frac{B}{2}) \sin (\frac{C}{2}) = \frac{4 abc}{4 ∆} \sqrt{\frac{(s - b) (s - c)}{bc}} \sqrt{\frac{(s - a) (s - c)}{ac}} \sqrt{\frac{(s - a) (s - b)}{ab}} = \frac{abc}{∆} \sqrt{\frac{{(s - b)}^{2} {(s - a)}^{2} {(s - c)}^{2}}{a^{2} b^{2} c^{2}}} = \frac{abc}{∆} \frac{∆^{2}}{s abc} = \frac{∆}{s} = r Hence, r = 4 Rsin (\frac{A}{2}) \sin (\frac{B}{2}) \sin (\frac{C}{2}) II . (s - a) \tan (\frac{A}{2}) = (s - a) \sqrt{\frac{(s - b) (s - c)}{s (s - a)}} = \sqrt{\frac{s (s - a) (s - b) (s - c)}{s^{2}}} = \frac{∆}{s} = r Hence, r_{1} \neq (s - a) \tan (\frac{A}{2}) Since, \frac{∆}{s - c} = \sqrt{\frac{s (s - a) (s - b)}{(s - c)}} = s \sqrt{\frac{(s - a) (s - b)}{s (s - c)}} = stan (\frac{c}{2}) = r_{3} Hence, r_{3} = \frac{∆}{s - c}$

Therefore I and III statements are true but II is a false statement.

829.

An aeroplane flying with uniform speed horizontally one km above the ground is observed at an elevation of 60°. After 10 s if the elevation is observed to be 30°, then the speed of the plane (in km/h) is