Extreme values and Periodicity of Trigonometric functions

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Question

The ratio of the greatest value of $2 - \cos x + \sin^{2} x$ is to its least value, is

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Solution

Let, $y = 2 - \cos x + \sin^{2} x$ then,
$\begin{aligned} y & = 2 - \cos x + 1 - \cos^{2} x \\ \Rightarrow y & = 3 - (\cos^{2} x + \cos x) \Rightarrow y = \frac{13}{4} - {(\cos x + \frac{1}{2})}^{2} \end{aligned}$
Now,
$\begin{aligned} - 1 \leq \cos x \leq 1 for all x \\ \Rightarrow & - \frac{1}{2} \leq (\cos x + \frac{1}{2}) \leq \frac{3}{2} for all x \\ \Rightarrow & 0 \leq {(\cos x + \frac{1}{2})}^{2} \leq \frac{9}{4} for all x \\ \Rightarrow & - \frac{9}{4} \leq - {(\cos x + \frac{1}{2})}^{2} \leq 0 for all x \\ \Rightarrow & \frac{13}{4} - \frac{9}{4} \leq \frac{13}{4} - {(\cos x + \frac{1}{2})}^{2} \leq \frac{13}{4} for all x \\ \Rightarrow & 1 \leq y \leq \frac{13}{4} \\ ∴ & y_{max} = \frac{13}{4} and y_{min} = 1 \end{aligned}$
Hence, required ratio is $\frac{13}{4}$

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Extreme values and Periodicity of Trigonometric functions

Remember concepts with our Masterclasses.

The ratio of the greatest value of 2−cos⁡x+sin2⁡x is to its least value, is

Ready to Test Your Skills?

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The ratio of the greatest value of $2 - \cos x + \sin^{2} x$ is to its least value, is