Trigonometrical Ratios And Identities Ques 18
Given $A=\sin ^{2} \theta+\cos ^{4} \theta$, then for all real values of $\theta$
(a) $1 \leq A \leq 2$
(b) $\frac{3}{4} \leq A \leq 1$
(c) $\frac{13}{16} \leq A \leq 1$
(d) $\frac{3}{4} \leq A \leq \frac{13}{16}$
$(1980,1 M)$
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Answer:
Correct Answer: 18.(b)
Solution:
Formula:
Domain and Range of Trigonometric Functions:
- Given, $A=\sin ^{2} \theta+\left(1-\sin ^{2} \theta\right)^{2}$
$ \begin{array}{ll} \Rightarrow & A=\sin ^{4} \theta-\sin ^{2} \theta+1 \\ \Rightarrow & A=(\sin ^{2} \theta-\frac{1}{2})^{2}+\frac{3}{4} \\ \Rightarrow & 0 \leq (\sin ^{2} \theta-\frac{1}{2})^{2} \leq \frac{1}{4} \quad\left[\because 0 \leq \sin ^{2} \theta \leq 1\right] \\ \therefore & \frac{3}{4} \leq A \leq 1 \end{array} $
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