Question

If $\tan x=\sqrt{3}$ , x lies in the first quadrant, find the value of the other five trigonometric functions.

Answer 1

Hint: Start by finding the value of sec x using the relation that the difference of squares of secant and tangent function is equal to 1. Now once we have got the value of secx, we can easily find other trigonometric ratios using the relation between the trigonometric ratios.

Complete step-by-step answer:
Before moving to the solution, let us discuss the sine and cosine function, which we would be using in the solution. All the trigonometric ratios, including sine and cosine, are periodic functions. We can better understand this using the graph of sine and cosine.
First, let us start with the graph of sinx.

Next, let us see the graph of cosx.

Similarly, we can draw the graphs of the other trigonometric ratios as well, and the graphs of the trigonometric ratios are very useful as well.
We will now start with the solution to the above question by finding the value of the secant function.
We know that ${{\sec }^{2}}x=1+{{\tan }^{2}}x.$ So, if we put the value of tanx in the formula, we get
$se{{c}^{2}}x=1+{{\left( \sqrt{3} \right)}^{2}}$
$\Rightarrow se{{c}^{2}}x=1+3$
$\Rightarrow {{\sec }^{2}}x=4$
Now we know that ${{a}^{2}}=b$ implies $a=\pm \sqrt{b}$ . So, our equation becomes:
$\Rightarrow \sec x=\pm \sqrt{4}=\pm 2$
Now, as it is given that x lies in the first quadrant and secx is positive in the first quadrant.
$\therefore secx=2$
Now we know that cosx is inverse of secx.
$\therefore \cos x=\dfrac{1}{\sec x}=\dfrac{1}{2}$
Now using the property that tanx is the ratio of sinx to cosx, we get
$\tan x=\dfrac{\sin x}{\cos x}$
$\Rightarrow \tan x\cos x=\sin x$
$\Rightarrow \sin x=\sqrt{3}\times \left( \dfrac{1}{2} \right)=\dfrac{\sqrt{3}}{2}$
Now we know that cotx is the inverse of tanx. So, we can conclude that:
$\cot x=\dfrac{1}{\tan x}=\dfrac{1}{\sqrt{3}}$
Also, we know that secx is the inverse of cosx, and cosecx is the inverse of sinx.
$\therefore \cos ecx=\dfrac{1}{\sin x}=\dfrac{2}{\sqrt{3}}$
$\therefore secx=\dfrac{1}{\cos x}=2$

Note: It is useful to remember the graph of the trigonometric ratios along with the signs of their values in different quadrants. For example: sine is always positive in the first and the second quadrant while negative in the other two. Also, you need to remember the properties related to complementary angles and trigonometric ratios.