Question

Given that $\cos (x) = - \dfrac{3}{4}\,$; what is the value of $x$?

Answer 1

Hint: The ratio between the base of the right-angled triangle and the hypotenuse of the right-angled triangle is called the cosine function. A right- angled triangle is the one that has equal to $90^\circ $. In this question we have to find the value of $x$ . It can be calculated by taking the inverse of the cosine function.

Complete step by step answer:
In this question we have to find the value of $x$ and it can be calculated by taking the inverse of the cosine function. Inverse cosine is one of the trigonometric functions. Each trigonometric function has an inverse of it, whether it is sine, cosine, tangent, secant, cosecant and cotangent. These functions are also widely used, apart from the trigonometric formulas, to solve many problems. Inverse functions are also called Arc functions because they give the length of the arc for a given value of trigonometric functions.

The inverse cosine function is the inverse of the cosine function and is used to obtain the values of angles for a right-angled triangle.Let $\cos A = 1$. Then,
$A = {\cos ^{ - 1}}1$
The value of cos and cos inverse is the same.
So $A = 0$
In the question we have given that $\cos (x) = - \dfrac{3}{4}\,$ we have to find the value of $x$.
So, $x = {\cos ^{ - 1}}\left( { - \dfrac{3}{4}} \right)$
We know that $\cos ( - x) = \cos x$
So, $x = {\cos ^{ - 1}}\left( {\dfrac{3}{4}} \right)$

Hence, the value of $x \approx \pm 2.41885806$ (in radians).

Note:$ - \dfrac{3}{4}$ is not one of the special values of the basic trigonometric functions. So, it is difficult to express $x$ exactly in this case. Some of the so-called “special angles” and their corresponding outputs for the cosine functions are as follows: $x = 0 \Rightarrow \cos x = 1$, $x = \dfrac{\Pi }{6} \Rightarrow \cos x = \dfrac{{\sqrt 3 }}{2}$.