Question

How do I find the value of sec 5pi/6 ?

Answer 1

Hint: Convert the angle to the required form to apply the ASTC rule. Express sec in terms of cos by taking the reciprocal. Then put the cos value of the angle and do the necessary calculation to obtain the required solution.

Complete step by step solution:
ASTC rule: we have different trigonometric functions like sin, cos, tan etc. ASTC stands for all, sin, tan, cos. This rule indicates the positivity of a particular trigonometric function on a particular quadrant as per the following table. For even multipliers of angle $\dfrac{\pi }{2}$ , the function remains the same. But for an odd multiplier of angle $\dfrac{\pi }{2}$ the values change accordingly.

Quadrant	Positive function
${{1}^{st}}$	All
${{2}^{nd}}$	sin and cosec
${{3}^{rd}}$	tan and cot
${{4}^{th}}$	cos and sec

We have, \[\sec \dfrac{5\pi }{6}\]
\[\sec \dfrac{5\pi }{6}\] can be written as \[\sec \dfrac{5\pi }{6}=\sec \left( \dfrac{\pi }{2}\times 2-\dfrac{\pi }{6} \right)\]
Since the angle is even multiplier of angle $\dfrac{\pi }{2}$, so the function remains the same i.e. sec.
And the angle lies in the ${{2}^{nd}}$ quadrant (as each quadrant is consists of $\dfrac{\pi }{2}$).
From the above table we can conclude that sec is a negative function in the ${{2}^{nd}}$ quadrant.
Now, our angle becomes
\[\sec \dfrac{5\pi }{6}=-\sec \dfrac{\pi }{6}\]
As we know, $\sec \theta =\dfrac{1}{\cos \theta }$
So, \[\sec \dfrac{\pi }{6}\] can be written as \[\sec \dfrac{\pi }{6}=\dfrac{1}{\cos \dfrac{\pi }{6}}\]
Putting the value of \[\sec \dfrac{\pi }{6}\], we get
 \[\sec \dfrac{5\pi }{6}=-\dfrac{1}{\cos \dfrac{\pi }{6}}\]
Again as we know \[\cos \dfrac{\pi }{6}=\dfrac{\sqrt{3}}{2}\]
Putting the value of \[\cos \dfrac{\pi }{6}\], we get
\[\sec \dfrac{5\pi }{6}=-\dfrac{1}{\dfrac{\sqrt{3}}{2}}=-1\times \dfrac{2}{\sqrt{3}}=-\dfrac{2}{\sqrt{3}}\]
Multiplying the numerator and the denominator with $\sqrt{3}$ we get
 \[\sec \dfrac{5\pi }{6}=-\dfrac{2\times \sqrt{3}}{\sqrt{3}\times \sqrt{3}}=-\dfrac{2\sqrt{3}}{3}\]
This is the required solution.

Note: Converting to the required form is the most important step, to which ASTC rule will be applicable. Some basic sin and cos angle values should be remembered for faster calculations.