No calculus, please

Algebra Level pending

y = tan ⁡ ( x + 2 π 3 ) − tan ⁡ ( x + π 6 ) + cos ⁡ ( x + π 6 ) ; − 5 π 12 ≤ x ≤ − π 3 y=\tan \left(x+\frac {2π}{3}\right) - \tan \left(x+\frac {π}{6}\right) + \cos \left(x+\frac {π}{6}\right); \quad -\frac {5π}{12}≤x≤-\frac {π}{3}

Find the minimum value of y y .


The answer is 2.707.

Parth Sankhe by Parth Sankhe , 27, India

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1 solution

Parth Sankhe
Oct 28, 2018

Substituting − x − π 6 = θ -x-\frac {π}{6}=\theta will give us y = cot ⁡ θ + tan ⁡ θ + cos ⁡ θ y=\cot\theta + \tan\theta + \cos\theta where π 6 ≤ θ ≤ π 4 \frac {π}{6}≤\theta≤\frac {π}{4} . y y is a decreasing function in this interval, thus minimum value will occur at θ = π 4 \theta = \frac {π}{4} . The min value is 2 + 1 √ 2 2+\frac {1}{√2} .

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You should put a backslash '\' before all function in Latex \cot \theta cot ⁡ θ \cot \theta , \tan \theta tan ⁡ θ \tan \theta and \cos \theta cos ⁡ θ \cos \theta . Note that the function names are not in italic which is for variables and constants. Also there is a space between the function and the variable θ \theta which is not there if you don't use a '\'. You can see the LaTex code by placing the mouse cursor on the formulas. I have edited your problem here.

Chew-Seong Cheong - 2 years, 7 months ago

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I'm sorry, will do next time. Thank you!

Parth Sankhe - 2 years, 7 months ago

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