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General Solution of Trigonometric Equation

The number of...

Question

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 $cos 2 θ = sin 4 θ$ is

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Solution

$tan θ = cot 5 θ \Rightarrow tan θ = tan (\frac{π}{2} - 5 θ) \Rightarrow θ = n π + \frac{π}{2} - 5 θ \Rightarrow θ = \frac{n π}{6} + \frac{π}{12}, n \in Z$
$∴ θ = {\pm \frac{π}{12}, \pm \frac{π}{4}, \pm \frac{5 π}{12}} \dots (i)$
Now $cos 2 θ = sin 4 θ$
$cos 2 θ = cos (\frac{π}{2} - 4 θ) \Rightarrow 2 θ = 2 n π \pm (\frac{π}{2} - 4 θ) \Rightarrow θ = \frac{(4 n + 1) π}{12}, - \frac{(4 n - 1) π}{4}, n \in Z ∴ θ = {- \frac{π}{4}, \frac{π}{12}, \frac{π}{4}, \frac{5 π}{12}} \dots (i i)$
from $(i)$ and $(i i)$
$θ = {- \frac{π}{4}, \frac{π}{12}, \frac{π}{4}, \frac{5 π}{12}}$

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

Q. The number of solution(s) of

cos 2 θ = sin 4 θ

tan θ = cot 5 θ

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. The possible values of

θ \in (0, π)

sin (θ) + sin (4 θ) + sin (7 θ) = 0

tan θ - i,

θ \in (- \frac{π}{2}, \frac{π}{2})

can be written as

r e^{i β}

(r, β)

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General Solution of Trigonometric Equation

Standard XII Mathematics

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