Question

How do you use a calculator to approximate $\arccos ( - 0.7)$ ?

Answer 1

Hint : “Arccos” stands for arccosine function. The arccosine function is the inverse function of $\cos (x).$ It can be calculated using formulas and graphs or even with a calculator. To solve it with the help of a calculator, we require scientific calculators as basic calculators cannot handle trigonometric calculations.

Complete step by step solution:
The arccosine of $x$ is defined as the inverse cosine function of $x$ when $- 1 \leqslant x \leqslant 1$ .
When the cosine of $y$ is equal to $x$ :
$\cos y = x$
Then the arccosine of $x$ is equal to the inverse cosine function of $x$ , which is equal to $y$ :
$\arccos \,x = {\cos ^{ - 1}}x = y$
For example, If the cosine of ${60^ \circ }$ is $0.5$ :
$\cos ({60^ \circ }) = 0.5$
Then the arccos of $0.5$ is ${60^ \circ }$ :
$\arccos (0.5) = {\cos ^{ - 1}}0.5 = {60^ \circ }$

We can proceed to calculate arccos in calculator as follows:
Step 1: Decide how you want your final answer to be displayed – whether in degrees or radiant and set the calculator mode accordingly.

Step 2: After deciding the mode, press $\cos$ on a calculator such that it calculates ${\cos ^{ - 1}}$ . We can generally access ${\cos ^{ - 1}}$ easily on a calculator by pressing a button called $2ND$ .

Step 3: Press the value for which you want to find the arccos. In the given case, press $- 0.7$ .

Step 4: Press Enter.
The calculator will display the final answer as per the pre-set format as follows:
In degrees the answer will be:
$\arccos ( - 0.7) = \pm {134^ \circ }42$
In radians the answer will be:
$\arccos ( - 0.7) = \pm 2.3462$

Note : Here ${\cos ^{ - 1}}x$ means the inverse cosine and does not mean cosine to the power of $- 1$ .
The formula for arccos of negative argument is as follows:
$\arccos ( - x) = \pi - \arccos \,x = {180^ \circ } - \arccos \,x$
Remember to set the mode on your calculator and only then proceed to solve the sum.