Question

How do you find the exact value in radians without using a calculator \[{\cos ^{ - 1}}\left( {\dfrac{1}{2}} \right)\]?

Answer 1

Hint: In this type of problem where we have to find value of trigonometric angles in degree or radians, firstly we try to convert inverse form into direct form and then try to find the angles related to given value in the problem and then convert either degree to radian or radian to degree as per requirement.

Formula used:
\[\cos {60^ \circ } = \dfrac{1}{2}\]
\[\pi = 3.14\]

Complete step by step answer:
Assuming value of \[{\cos ^{ - 1}}\left( {\dfrac{1}{2}} \right)\] in radians be ‘\[\theta \]’, we get
\[ \Rightarrow \theta = {\cos ^{ - 1}}\left( {\dfrac{1}{2}} \right)\]
Now, taking ‘\[\cos \]’ both sides, we get
\[ \Rightarrow \cos \theta = \cos \left[ {{{\cos }^{ - 1}}\left( {\dfrac{1}{2}} \right)} \right]\]
At right hand side ‘\[\cos \]’ and ‘\[{\cos ^{ - 1}}\]’ will get cancel out, so we get the following equation
\[ \Rightarrow \cos \theta = \dfrac{1}{2}\]
Now, replacing \[\dfrac{1}{2}\]with ‘\[\cos {60^ \circ }\]’ by using above given formula, we get
\[ \Rightarrow \cos \theta = \cos {60^ \circ }\]
‘\[\cos \]’ will get cancel out from both sides, we get
\[ \Rightarrow \theta = {60^ \circ }\]
Now, converting the value of angle which is in degree to radian. To convert it we are multiplying right hand side with \[\dfrac{\pi }{{{{180}^ \circ }}}\], we get
\[ \Rightarrow \theta = {60^ \circ } \times \dfrac{\pi }{{{{180}^ \circ }}}\]
\[{180^ \circ }\] will get cancel out by \[{60^ \circ }\]with three times, we get
\[ \Rightarrow \theta = \dfrac{\pi }{3}\]
Putting value of \[\pi = 3.14\], we get
\[ \Rightarrow \theta = \dfrac{{3.14}}{3}\]
Now, dividing 3.14 by 3, we get
\[ \Rightarrow \theta = 1.04\]

So, we got the final value of \[{\cos ^{ - 1}}\left( {\dfrac{1}{2}} \right) = 1.04\]in radian.

Note: In these types of trigonometric angles-based problems value of some important angles of trigonometric ratios such that \[\sin {0^ \circ },\sin {30^ \circ },\sin {45^ \circ },\sin {60^ \circ },\sin {90^ \circ },\cos {0^ \circ },\cos {30^ \circ },\cos {45^ \circ },\]\[\cos {60^ \circ },\cos {90^ \circ }\]should be known so that we can easily convert it from inverse form to direct form. The second thing is conversion of degree into radian or radian into degree. If we are going to convert degree into radian, we have to multiply the given value to \[\pi /{180^ \circ }\] and if we have to convert from radian to degree, we have to multiply the given value to \[{180^ \circ }/\pi \].