Question

How do you find the exact value of $\sin x$ when $\cos x = \dfrac{3}{5}$ and the terminal side of $x$ is in quadrant $4$ ?

Answer 1

For solving this particular question , you have to simplify the given expression by reordering the equation , applying trigonometric identity , taking square root. To find the exact value of $\sin x$ , we must know that ${\sin ^2}x + {\cos ^2}x = 1$ .

Complete step by step solution:
We have to find the exact value of
$\sin (x)$
We have given ,
$\cos x = \dfrac{3}{5}$
Taking square root both the side of the above equation, we will get ,
$\Rightarrow {\cos ^2}x = \dfrac{9}{{25}}.........(1)$
For finding $\sin (x)$ , we must use the identity ${\sin ^2}x + {\cos ^2}x = 1$ ,
$\Rightarrow {\sin ^2}x + {\cos ^2}x = 1$
Substitute $(1)$ in the above equation ,
$\Rightarrow {\sin ^2}x + \dfrac{9}{{25}} = 1$
Subtract $\dfrac{9}{{25}}$ from both the side ,
$\Rightarrow {\sin ^2}x = 1 - \dfrac{9}{{25}} \\\ \Rightarrow {\sin ^2}x = \dfrac{{16}}{{25}} \\\$
Taking square root both the side ,
$\Rightarrow \sin x = \pm \dfrac{4}{5}$
$\Rightarrow \sin x = \dfrac{4}{5}$ is the only solution , since it is in the fourth quadrant.

Therefore , the exact value of $\sin x = \dfrac{4}{5}$ .

Additional Information:
In arithmetic, pure mathematics identities are equalities that involve pure mathematics functions and are true for every worth of the occurring variables that every aspect of the equality is outlined. Geometrically, these are identities involving sure functions of one or additional angles. We have a trigonometry formula which says $\sin \theta = \dfrac{p}{h}$ , where $p$ represent length of perpendicular side and $h$represent length of hypotenuse side. We have another trigonometry formula which says $\cos \theta = \dfrac{b}{h}$ , where $b$ represents length of base and $h$ represent length of hypotenuse side.

Note: Questions similar in nature as that of above can be approached in a similar manner and we can solve it easily. In order to solve and simplify the given expression we can also use the identity that is ${\sin ^2}x + {\cos ^2}x = 1$ and express our given expression in the simplest form and thereby solve it. Identities are helpful whenever expressions involving pure mathematics functions should be simplified.