Question

How do you write the trigonometric function $\sec x$ in terms of $\sin x?$

Answer 1

Hint: Here we are going to use the inverse trigonometric functions relating to secant and cosine. Also, we will use the correlation stating for sine and cosine and then ultimately will derive equations correlating secant and sine angles as per requirement.

Complete step-by-step solution:
Here we will use the reciprocal identity.
Secant and cosine are inverse of each other. Therefore, it can be expressed as –
$\Rightarrow \sec x = \dfrac{1}{{\cos x}}$
Also, by using Pythagorean identity-
$\Rightarrow \cos x = \sin \left( {\dfrac{\pi }{2} - x} \right)$
Also, it can also be expressed as –
$\Rightarrow \cos x = \sqrt {1 - {{\sin }^2}x} $
Hence, the required solution is-
$\Rightarrow \sec x = \dfrac{1}{{\sin \left( {\dfrac{\pi }{2} - x} \right)}}$ or
$\Rightarrow \sec x = \dfrac{1}{{\sqrt {1 - {{\sin }^2}x} }}$

Additional Information: Cosine and secant are inverse functions of each other and sine and cosecant are the inverse functions of each other. 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 $si{n^2}\theta + co{s^2}\theta = 1$ and derive other trigonometric functions using it such as tan, cosec, cot and cosec angles.

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 $ ).

Note: Always remember the correlation between the trigonometric functions and the angles. Also, know the Pythagorean theorem for the angles to get the equivalent values for the different angles and different trigonometric functions.