The number of values of - in the interval -2- -2 such that - n-5 for n -0- - 1- - 2 and tan- 5 - as well as sin 2 -cos 4 - is

Given, tanθ =cot5θ ⇒ tanθ=tan ( π/2 5θ ) ⇒π/2 5θ=nπ+θ ⇒ 6θ=π/2 nπ ⇒θ=π/12 nπ/6 Also cos4θ=sin2θ=cos ( π/2 2θ ) ⇒ 4θ=2nπ± ( π/2 2θ ) Taking positive sign, 6θ=2nπ ...

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Standard XI

Mathematics

Trigonometric Ratios of Compound Angles

The number of...

Question

The number of values of $θ$ in the interval $(- \frac{π}{2}, \frac{π}{2})$ such that $θ \neq \frac{n π}{5}$ for $n = 0, \pm 1, \pm 2$ and $t a n θ = c o t 5 θ$ as well as $s i n 2 θ = c o s 4 θ$ is ___

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Solution

Given, $t a n θ = c o t 5 θ$
$\Rightarrow t a n θ = t a n (\frac{π}{2} - 5 θ) \Rightarrow \frac{π}{2} - 5 θ = n π + θ \Rightarrow 6 θ = \frac{π}{2} - n π \Rightarrow θ = \frac{π}{12} - \frac{n π}{6}$
Also $c o s 4 θ = s i n 2 θ = c o s (\frac{π}{2} - 2 θ)$
$\Rightarrow 4 θ = 2 n π \pm (\frac{π}{2} - 2 θ)$
Taking positive sign,
$6 θ = 2 n π + \frac{π}{2} \Rightarrow θ = \frac{n π}{3} + \frac{π}{12}$
Taking negative sign,
$2 θ = 2 n π - \frac{π}{2} \Rightarrow θ = n π - \frac{π}{4}$
Also values of $θ$ suggest that there are only 3 common solutions in the interval of $(- \frac{π}{2}, \frac{π}{2})$ .

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Similar questions

Q. The number of values of

θ

in the interval

(- \frac{π}{2}, \frac{π}{2})

such that

tan θ = cot 5 θ

as well as

cos 2 θ = sin 4 θ

Q. The number of solution(s) of

cos 2 θ = sin 4 θ

and

tan θ = cot 5 θ

for

θ \in (- \frac{π}{2}, \frac{π}{2})

Some trigonometric ratios and the interval in which $θ$ lies is given. Match the intervals with the ratios which are positive in those intervals.

$θ$ gives positive values

p. (0, $\frac{π}{2}$ ) 1. Only sin $θ$ , cosec $θ$

q. ( $\frac{π}{2}$ , $π$ ) 2. Only cos $θ$ , sec $θ$

r. $(π, \frac{3 π}{2})$ 3. Only tan $θ$ , cot $θ$

s. $(\frac{3 π}{2}, 2 π)$ 4. All sin $θ$ , cos $θ$ , tan $θ$ , cot $θ$ , sec $θ$ , cosec $θ$

Q. Find all values of

θ

in the interval

(- π / 2, π / 2)

satisfying the equation

(1 - t a n θ) (1 + t a n θ) s e c^{2} θ + 2^{t a n^{2} θ} = 0

Q. A value of

θ

in the interval

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satisfying the equation

(1 - t a n θ) (1 + t a n θ) s e c^{2} θ + 2 t a n^{2} θ + 2 = 0

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The number of values of θ in the interval (−π2,π2) such that θ≠nπ5 for n=0,±1,±2 and tanθ=cot5θ as well as sin2θ=cos4θ is ___

The number of values of $θ$ in the interval $(- \frac{π}{2}, \frac{π}{2})$ such that $θ \neq \frac{n π}{5}$ for $n = 0, \pm 1, \pm 2$ and $t a n θ = c o t 5 θ$ as well as $s i n 2 θ = c o s 4 θ$ is ___