Question

How do you find $\sin$ if $\tan$ is 4?

Answer 1

In order to determine the sine when tangent of some angle $x$ is 4 , put $\tan = 4$ in the identity ${\sec ^2}x = {\tan ^2}x + 1$ to find the value of ${\cos ^2}x$ and put this value of ${\cos^2}x$ in the identity of trigonometry ${\sin ^2}x + {\cos ^2}x = 1$ to determine the value of $\sin x$

Complete step by step solution:
We are given that the tangent of some $x$ is equal to $4$
$\tan x = 4$
As we know the identity of trigonometry that the sum of tangent square and one is equal to the square of secant.
${\sec ^2}x = {\tan ^2}x + 1$
Putting $\tan x = 4$,we get
$\Rightarrow {\sec ^2}x = {\left( 4 \right)^2} + 1 \\\ \Rightarrow {\sec ^2}x = 16 + 1 \\\ \Rightarrow {\sec ^2}x = 17$
Secant is nothing but the reciprocal of cosine $\sec x = \dfrac{1}{{\cos x}}$
$\Rightarrow \dfrac{1}{{{{\cos }^2}x}} = 17$
Taking reciprocal on both sides of the equation , we get
$\Rightarrow {\cos ^2}x = \dfrac{1}{{17}}$
Now putting the above value in the identity of trigonometry ${\sin ^2}x + {\cos ^2}x = 1$, we get
$\Rightarrow {\sin ^2}x + \dfrac{1}{{17}} = 1 \\\ \Rightarrow {\sin ^2}x = 1 - \dfrac{1}{{17}} \\\ \Rightarrow {\sin ^2}x = \dfrac{{17 - 1}}{{17}} \\\ \Rightarrow {\sin ^2}x = \dfrac{{16}}{{17}}$
Taking square root on both sides of the equation,
$\Rightarrow \sin x = \dfrac{{\sqrt {16} }}{{\sqrt {17} }}$ $\sqrt {16} = 4$
$\therefore \sin x = \dfrac{4}{{\sqrt {17} }}$

Therefore, the value of $\sin x$ is equal to $\dfrac{4}{{\sqrt {17} }}$ when $\tan x = 4$.

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. We know that $\sin ( - \theta ) = - \sin \theta .\cos ( - \theta ) = \cos \theta \,and\,\tan ( - \theta ) = - \tan \theta$. Therefore,$\sin \theta$ and $\tan \theta$ and their reciprocals, $cosec \theta$ and $\cot \theta$ are odd functions whereas $\cos \theta$ and its reciprocal $\sec \theta$ are even functions.
4. Periodic Function= A function $f(x)$ is said to be a periodic function if there exists a real number T > 0 such that $f(x + T) = f(x)$ for all x.

Note: One must be careful while taking values from the trigonometric table and cross-check at least once to avoid any error in the answer.Use the identities carefully.As secant is reciprocal of cosine, also cosecant is reciprocal of sine and cotangent is reciprocal of tangent.