Question

Period of $ \left| {\sin x} \right| + \left| {\cos x} \right| $ is
A. $ \dfrac{\pi }{2} $
B. $ \pi $
C. $ \dfrac{{3\pi }}{2} $
D. $ 2\pi $

Answer 1

Hint: Split the given functions into two terms. Here we will find the period of the given functions of the two terms one by one and then take its LCM (least common multiple) with the average of both the two periods.

Complete step-by-step answer:
Take the given function:
Split both the terms. We know that the sine and cosine have a period of $ \pi {\text{ or 180}}^\circ $ means the values of the sine and cosine are repeated over the period.
Let, $ f(x) = \left| {\sin x} \right| $
Period of $ \left| {\sin x} \right| = \pi $ ..... (i)
Also, let us suppose that $ g(x) = \left| {co\operatorname{s} x} \right| $
Period of $ \left| {\cos x} \right| = \pi $ ..... (ii)
Therefore, period of the given function –
 $ \left| {\sin x} \right| + \left| {\cos x} \right| = \dfrac{1}{2}(LCM{\text{ of }}\pi {\text{,}}\pi {\text{)}} $
\[
   \Rightarrow \left| {\sin x} \right| + \left| {\cos x} \right| = \dfrac{1}{2}(\pi {\text{)}} \\
   \Rightarrow \left| {\sin x} \right| + \left| {\cos x} \right| = \dfrac{\pi }{2} \\
 \]
Hence, from the given multiple choices – the option A is the correct answer.
So, the correct answer is “Option A”.

Note: Also, remember that the most important property of sines and cosines is that their values lie between minus one and plus one. Every point on the circle is unit circle from the origin. So, the coordinates of any point are within one of zero as well.
Directly the Pythagoras identity are followed by sines and cosines which concludes that: $ {\operatorname{Sin} ^2}\theta + {\operatorname{Cos} ^2}\theta = 1 $
Remember the trigonometric formulas and the correlation between the trigonometric functions to find the unknowns. Also, remember the All STC rule, it is also known as the ASTC rule in geometry. It states that all the trigonometric ratios in the first quadrant ( $ 0^\circ \;{\text{to 90}}^\circ $ ) are positive, sine and cosec are positive in the second quadrant ( $ 90^\circ {\text{ to 180}}^\circ $ ), tan and cot are positive in the third quadrant ( $ 180^\circ \;{\text{to 270}}^\circ $ ) and sin and cosec are positive in the fourth quadrant ( $ 270^\circ {\text{ to 360}}^\circ $ ).