Question

How do you find the value of the expression \[\arccos \left( { - 0.7} \right)\]?

Answer 1

Hint: In the above question, is based on the inverse trigonometry concept. The trigonometric functions are the relationship between the angles and the sides of the triangle. Since measure is given in the function, we need to find the angle of that particular measure of trigonometric function.

Complete step-by-step solution:
Given expression: \[\arccos \left( { - 0.7} \right)\]
$\arccos $ is an inverse trigonometric function which can also be written as ${\cos ^{ - 1}}$. $ - 1$ here is just the way of showing that it is inverse of $\cos x$. Inverse cosine does the opposite of cosine. Cosine function gives the angle which is calculated by dividing the adjacent side and hypotenuse in a right-angle triangle, but the inverse of it gives the measure of an angle.
The cosine of angle $\theta $ is:
$\cos \theta = \dfrac{{Adjacent}}{{Hypotenuse}}$
Therefore, inverse of cosine is:
${\cos ^{ - 1}}\left( {\dfrac{{Adjacent}}{{Hypotenuse}}} \right) = \theta $
In the above cosine function, we have to find inverse of cosine function with given value $ - 0.7$
\[\arccos \left( { - 0.7} \right) = {134^ \circ }\]
Let $x = \arccos \left( { - 0.7} \right)$
We have to find all possible values of $x$ satisfying the equation $x = \arccos \left( { - 0.7} \right)$.
So, take $\cos $ on both sides of the equation to extract $x$ from inside the $\cos $.
$\cos x = - 0.7$
Now, we will find the values of $x$ satisfying $\cos x = - 0.7$…(i)
So, using the property $\cos \left( {\pi - x} \right) = - \cos x$ and $\cos \left( {0.7} \right) = 0.7953988302$ in equation (i).
$ \Rightarrow \cos x = - \cos \left( {0.795} \right)$
$ \Rightarrow \cos x = \cos \left( {3.14 - 0.795} \right)$
$ \Rightarrow x = 2.345$
Now, using the property $\cos \left( {\pi + x} \right) = - \cos x$ and $\cos \left( {0.7} \right) = 0.7953988302$ in equation (i).
$ \Rightarrow \cos x = - \cos \left( {0.795} \right)$
$ \Rightarrow \cos x = \cos \left( {3.14 + 0.795} \right)$
$ \Rightarrow x = 3.935$
Since, the period of the $\cos \left( x \right)$ function is $2\pi $ so values will repeat every $2\pi $ radians in both directions.
$x = 2.345 + 2n\pi ,3.935 + 2n\pi $, for any integer $n$.
Hence, the value of the expression \[\arccos \left( { - 0.7} \right)\] is $134$ degrees and $2.345$ radians.

Note: An important thing to note is that since the value is a negative value the cosine function is positive in the fourth quadrant so it is easy to predict then angle will be between 0 to 360 degree. If you need to find out the value in the first quadrant, we can subtract $\pi $ by 3.14 since the period of cosine is $2\pi $.