Question

How do you evaluate \[Arc\,\cos \left( {\dfrac{4}{5}} \right)\] ?

Answer 1

Hint: To solve this question, you should know about inverse trigonometric functions.
Trigonometric functions are sin, cos, tan, cosec, cot and sec. Finding the inverse value for these functions are called as Inverse trigonometric functions. In this given case you have to first solve the fraction and then have to find the closest angle in the unit circle which gives the value that has been given.

Complete step by step answer:
Every mathematical function, from the simplest to the most complex, has an inverse.
In mathematics, inverse usually means opposite.
 For addition, the inverse is subtraction.
 For multiplication, it's division.
And for trigonometric functions, it's the inverse trigonometric functions.
The inverse trigonometric functions are used to determine the angle measure when at least two sides of a right triangle are known.
So now for the given question we have to find the inverse of cos
Therefore we can write it as
\[ \Rightarrow \cos \theta = \dfrac{4}{5}\]
So arc cos can be found by
$ \Rightarrow \operatorname{arc} \,\cos \theta = \dfrac{4}{5}$
When we solve the fraction we get
$ \Rightarrow arc\,\cos \theta = 0.8$
$ \Rightarrow arc\,\theta = {\cos ^{ - 1}}\left( {0.8} \right)$
So in the unit circle the closest value of $0.8$in cos function is
$ \Rightarrow arc\theta = {36^ \circ }87$
Hence the inverse function for cos is found.

Note:
Misconception you will have while solving this problem:
$1)$ The expression ${\sin ^{ - 1}}\left( x \right)$ is not the same as $\dfrac{1}{{\sin x}}$. In other words, the $ - 1$ is not an exponent. Instead, it simply means inverse function.
$2)$ While solving for some angle you should be very careful about the quadrant in which the angle is present. Because the sign of the angle is one main criteria that gives you the correct solution
$3)$ We can also express the inverse sine as $\arcsin $and the inverse cosine as $\arccos $ and the inverse tangent as$\arctan $. They both represent the inverse trigonometric functions only. This notation is common in computer programming languages, and less common in mathematics.