Question
Question: How do you solve \(\tan (x) = \dfrac{1}{2}\)?...
How do you solve tan(x)=21?
Solution
To solve the given question means to find the value of x. In order to find x here, we will use inverse trigonometric functions. The inverse trigonometric functions are the inverse functions of the trigonometric functions which are used to obtain an angle from any of the angle’s trigonometric ratios. Here we will be using an inverse tan function to find the solution.
Complete step by step solution:
The trigonometric equation is tan(x)=21.
On applying an inverse tan function, we get
⇒tan−1(tanx)=tan−1(21)
⇒x=tan−1(21)
⇒x=26.565o
Hence the solution of tan(x)=21 is x=26.565o.
Note: The domain of the inverse tan function is (−∞,∞)which means domain contains all the real numbers and the range is (−2π,2π) which means range contains all the angles between −2π and 2π but not −2π and 2π. In the question given above, when we used tan−1, the domain for it was tan(x). The question itself states that tan(x)=21, hence it passed the domain condition for the inverse tan function. Also tan−1(tanx)=x only when x∈(−2π,2π). Had xnot assumed and lied between (−2π,2π), we would not have been able to use tan−1(tanx)=x.
One more thing that should also be kept in mind while using inverse trigonometric function (specifically inverse tan function for the context given) is that the expressiontan−1(x) is not equal to tan(x)1, which means −1 is not an exponent of tan(x) here.