Number of roots of equation cos ^7 θ-sin ^4 θ=1 that lie in interval [0,2 π] is

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Trigonometrical Equations

hard

The number of roots of the equation $\cos ^7 \theta-\sin ^4 \theta=1$ that lie in the interval $[0,2 \pi]$ is

A

$2$

B

$3$

C

$4$

D

$8$

(KVPY-2010)

Solution

(a)

We have, $\cos ^7 \theta-\sin ^4 \theta=1$

$\Rightarrow \quad \cos ^7 \theta=1+\sin ^4 \theta$

LHS $\cos ^7 \theta \in[-1,1]$

$RHS \geq 1$

Hence, $\cos ^7 \theta=1$ and $\sin ^4 \theta=0$ $\therefore \theta=0,2 \pi$ in $[0,2 \pi]$

Standard 11

Mathematics

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