Question

What is $\arccos (0.921)$?

Answer 1

Generally in Mathematics, the inverse trigonometric functions are known as arcus function which is the inverse function of the trigonometric function. In particular, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions. The most common notation to denote these inverse trigonometric functions are by prefixing an arc before the trigonometric function (i.e.)$arcsin\left( x \right)$ ,$arccos\left( x \right)$ , $arctan\left( x \right)$,${\text{ }}arc{\text{ }}sec\left( x \right)$ and so here, our question is to find $arccos\left( x \right)$. As we discussed earlier, $arccos\left( x \right)$ can also be written as ${\cos ^{ - 1}}x$.

Complete step by step answer:
The$arccos\left( x \right)$is an inverse function of the cosine of $x$ when $x$ lies between $- 1$and $1$ (i.e.) $- 1 \leqslant x \leqslant 1$. When $\cos y$ is equal to $x$ (i.e.) $\cos y = x$. Then the arccosine of $x$ is equal to the inverse function of the cosine of $x$, that is equal to $y$.
$arccos\left( x \right) = {\cos ^{ - 1}}x = y$

Here, the inverse cosine function does not mean cosine to the power of $- 1$.From the above explanation, we can easily understand that $arccos\left( x \right)$can be found by looking at the value of $x$ in the logarithmic tables. That is, we need to look at the value of $x$i n the natural cosine table, and then our required answer will be obtained.

Now, let us move on to our question.We need to find the value of $\arccos (0.921)$.Here, $x$ is $0.921$. Now, let us look at the value of $0.921$ in the natural cosine table. Then,
$\arccos (0.921) = 22^\circ {55^1}$
We need to convert the above degrees and minutes into decimal format. Now,
${55^1} = {\left( {\dfrac{{55}}{{60}}} \right)^\circ }$
Just divide the value of minutes by $60$so that we can obtain the decimal form.
And then, add ${22^\circ }$with ${\left( {\dfrac{{55}}{{60}}} \right)^\circ }$.
${22^\circ } + {55^1} = {22^\circ } + {\left( {\dfrac{{55}}{{60}}} \right)^\circ }$
$\Rightarrow {22^\circ } + {55^1} = {\left( {22.92} \right)^\circ }$

Hence, $\arccos (0.921) = 22^\circ {55^1} = {\left( {22.92} \right)^\circ }$.

Note: The $arccos\left( x \right)$can be found by looking at the value of $x$ in the logarithmic tables. We can also ask to calculate the value of $arcsin\left( x \right)$, $arctan\left( x \right)$. Similarly, the values of remaining inverse trigonometric functions such as $arcsin\left( x \right)$, $arctan\left( x \right)$ can be calculated by using the above method.