Question

f(θ)=sinθ+cos(θ+3/π​).find least value​

Answer 1

Answer:

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Answer 2

Answer:

Step-by-step explanation:

We have,

$\tt{f(\theta)=sin(\theta)+cos\bigg(\theta+\dfrac{\pi}{3}\bigg)}$

$\sf{\implies\,f(\theta)=sin(\theta)+cos(\theta)cos\bigg(\dfrac{\pi}{3}\bigg)- sin(\theta)sin\bigg(\dfrac{\pi}{3}\bigg)}$

$\sf{\implies\,f(\theta)=sin(\theta)+\dfrac{1}{2}\cdot\,cos(\theta)- \dfrac{\sqrt{3}}{2}\cdot\,sin(\theta)}$

$\sf{\implies\,f(\theta)=\bigg(1-\dfrac{\sqrt{3}}{2}\bigg)\,sin(\theta)+\dfrac{1}{2}\cdot\,cos(\theta)}$

So, the least value will be,

$\sf{\sqrt{\bigg(1-\dfrac{\sqrt{3}}{2}\bigg)^2+\bigg(\dfrac{1}{2}\bigg)^2}}$

$\sf{=\sqrt{1+\dfrac{3}{4}-\sqrt{3}+\dfrac{1}{4}}}$

$\sf{=\sqrt{1+1-\sqrt{3}}}$

$\sf{=\sqrt{2-\sqrt{3}}}$