Question

How do you find the general solutions for $\cos x = 0.6$

Answer 1

Hint:In order to find the general solution of the above equation , we will take the inverse of cosine on both sides and consider the fact that $\cos ( - \theta ) = \cos \theta $. So the value of $x$ becomes $\pm {\cos ^{ - 1}}\left( {0.6} \right)$. As we know the graph of cosine repeats after every $2\pi $ interval ,so the general solution is$2n\pi \pm {\cos ^{ - 1}}\left( {0.6} \right)$, where n is any integer number.

Complete step by step solution:
We are given a trigonometric function $\cos x = 0.6$
Taking inverse of cosine on both sides of the equation ,we get
$
\cos x = 0.6 \\
{\cos ^{ - 1}}\left( {\cos x} \right) = {\cos ^{ - 1}}\left( {0.6} \right) \\
x = {\cos ^{ - 1}}\left( {0.6} \right) \\
$
As we know $\cos ( - \theta ) = \cos \theta $
So $x$ is also $ - {\cos ^{ - 1}}\left( {0.6} \right)$
The graph of cosine repeat after $2\pi $
Therefore the general solution to $\cos x = 0.6$ is equal to $2n\pi \pm {\cos ^{ - 1}}\left( {0.6} \right)$, where n is any integer number.

Additional Information:
1. Trigonometry is one of the significant branches throughout the entire existence of mathematics and this idea is given by a Greek mathematician Hipparchus.
2. Even Function – A function $f(x)$ is said to be an even function ,if $f( - x) = f(x)$for all x in its domain.
Odd Function – A function $f(x)$ is said to be an even function ,if $f( - x) = - f(x)$for all x in its domain..
3. In Mathematics the inverse trigonometric functions (every so often additionally called anti- trigonometric functions or cyclomatic function) are the reverse elements of the mathematical functions In particular, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are utilized to get a point from any of the point's mathematical proportions.
Reverse trigonometric functions are generally utilized in designing, route, material science, and calculation.
4. In inverse trigonometric function, the domain are the ranges of corresponding trigonometric functions and the range are the domain of the corresponding trigonometric function.

Note:
1.One must be careful while taking values from the trigonometric table and cross-check at least once to avoid any error in the answer.
2.Fromula should be correctly used at every point.