Question

How do you find $\sin \left( \dfrac{-pi}{2} \right)$ ?

Answer 1

Hint: Using trigonometric functions, we have to apply the trigonometric ratios for the particular angle and find its value. We should know about the trigonometric ratios for different angles. Also, we should know about the even and odd functions. In this question, particularly sine function is used.

Complete step by step answer:
Basic trigonometric functions are:
$\Rightarrow $ Sine (sin)
$\Rightarrow $Cosine (cos)
$\Rightarrow $Tangent (tan)
When we say $\sin \theta $, here $\theta $ means angle in degrees.
Derived functions are:
$\Rightarrow $cosec$\theta $ = $\dfrac{1}{\sin \theta }$
$\Rightarrow $sec$\theta $ = $\dfrac{1}{\cos \theta }$
$\Rightarrow $tan$\theta $ = $\dfrac{\sin \theta }{\cos \theta }$ = $\dfrac{1}{\cot \theta }$
$\Rightarrow $cot$\theta $ = $\dfrac{1}{\tan \theta }$ = $\dfrac{\cos \theta }{\sin \theta }$
You know what is sin$\theta $? Let’s find out.

So, from figure, sin$\theta $ = $\dfrac{perpendicular(P)}{hypotenuse(H)}$
Now, let’s see some even and odd functions.
$\Rightarrow $sin(-x) = -sinx
$\Rightarrow $ cos(-x) = cosx
$\Rightarrow $ tan(-x) = -tanx
$\Rightarrow $ cot(-x) = -cotx
$\Rightarrow $ cosec(-x) = -cosecx
$\Rightarrow $sec(-x) = secx
Now, let’s make a table of trigonometric ratios for basic trigonometric functions i.e. sin, cos, tan, cot, sec and cosec.

Trigonometric ratios(angle $\theta $ in degrees)	${{0}^{\circ }}$	${{30}^{\circ }}$	${{45}^{\circ }}$	${{60}^{\circ }}$	${{90}^{\circ }}$
sin$\theta $	0	$\dfrac{1}{2}$	$\dfrac{1}{\sqrt{2}}$	$\dfrac{\sqrt{3}}{2}$	1
cos$\theta $	1	$\dfrac{\sqrt{3}}{2}$	$\dfrac{1}{\sqrt{2}}$	$\dfrac{1}{2}$	0
tan$\theta $	0	$\dfrac{1}{\sqrt{3}}$	1	$\sqrt{3}$	$\infty $
cosec$\theta $	$\infty $	2	$\sqrt{2}$	$\dfrac{2}{\sqrt{3}}$	1
sec$\theta $	1	$\dfrac{2}{\sqrt{3}}$	$\sqrt{2}$	2	$\infty $
cot$\theta $	$\infty $	$\sqrt{3}$	1	$\dfrac{1}{\sqrt{3}}$	0

So, from the trigonometric ratio table, we can see that the value of sin$\theta $ at ${{90}^{\circ }}$is 1.
As we can see in the table, values are given in degrees, but we need the value in radians.
So let’s convert ${{90}^{\circ }}$ to radians.
If we talk about a circle, so 1 revolution about a circle in degrees is ${{360}^{\circ }}$but in radians, it is $2\pi $ radians. So, they both are equal.
$\Rightarrow {{360}^{\circ }}=2\pi $radians
Now, let’s divide $2\pi $ by ${{360}^{\circ }}$:
$\Rightarrow 1=\dfrac{2\pi }{{{360}^{\circ }}}$
$\Rightarrow 1=\dfrac{\pi }{{{180}^{\circ }}}$
Basically, now we will plug in the angle we want to convert units with. So we will multiply with ${{90}^{\circ }}$.
$\Rightarrow \dfrac{\pi }{{{180}^{\circ }}}\times {{90}^{\circ }}$
After cancellation we will get:
$\dfrac{\pi }{2}$ rad
Now, we can say if $\sin \dfrac{\pi }{2}=1$ from the table, then $\sin \left( \dfrac{-\pi }{2} \right)=-1$ by using the even and odd function here:
$\Rightarrow $sin(-x) = -sinx
So, the final answer is: $\sin \left( \dfrac{-\pi }{2} \right)=-1$.

This is the graph of $\sin \left( \dfrac{-\pi }{2} \right)=-1$.

Note: Students must know the conversion of degrees to radians. You should remember all the functions and trigonometric ratios before solving any question related to trigonometry. Formula for conversion of degrees to radians is:
Radians = degrees$\times \dfrac{\pi }{{{180}^{\circ }}}$