Question

If $\cos \theta = - \dfrac{4}{9}$ with $\theta $ in quadrant 2, find $\sin \theta $?

Answer 1

Hint: We are given the value of $\cos \theta $. So, to find the value of $\sin \theta $ we will use the most fundamental formula of trigonometry that states ${\sin ^2}\theta + {\cos ^2}\theta = 1$. Replacing the value of $\cos \theta $ in the formula and then operating accordingly we will get the value of $\sin \theta $. Also, given that the angle $\theta $ lies in the second quadrant. So, we will have to simplify that more accordingly. So, let us see how to solve the problem.

Complete step-by-step answer:
Given, to us is $\cos \theta = - \dfrac{4}{9}$.
Now, we know the most basic relation between sine and cosine function is ${\sin ^2}\theta + {\cos ^2}\theta = 1$.
Therefore, now substituting the given value in the formula, we get,
${\sin ^2}\theta + {\left( { - \dfrac{4}{9}} \right)^2} = 1$
Now, on opening the brackets and squaring, we get,
$ \Rightarrow {\sin ^2}\theta + \dfrac{{16}}{{81}} = 1$
Now, subtracting $\dfrac{{16}}{{81}}$ both sides, we get,
$ \Rightarrow {\sin ^2}\theta = 1 - \dfrac{{16}}{{81}}$
Now, taking the LCM on right hand side, we get,
$ \Rightarrow {\sin ^2}\theta = \dfrac{{81 - 16}}{{81}}$
$ \Rightarrow {\sin ^2}\theta = \dfrac{{65}}{{81}}$
Now, taking square root on both the sides of the equation, we get,
$ \Rightarrow \sin \theta = \pm \sqrt {\dfrac{{65}}{{81}}} $
$ \Rightarrow \sin \theta = \pm \dfrac{{\sqrt {65} }}{9}$
Now, it is given to us in the question that the angle $\theta $ lies in the second quadrant.
We know, the value of $\sin \theta $ in the second quadrant is always positive.
Therefore, we will take only the positive value of $\sin \theta $.
Therefore, we can conclude that,
$\sin \theta = \dfrac{{\sqrt {65} }}{9}$

Note: The value of all the trigonometric functions are repetitive in the Cartesian plane because they are periodic in nature. Among the four quadrants, all the trigonometric functions are positive in the first quadrant. In the second quadrant only sine and cosecant functions are positive. In the third quadrant tangent and cotangent functions are positive. And in the fourth quadrant cosine and secant functions are positive.