Question

Why is \[\cos(\pi)\] and \[\cos( -\ pi)\] both equal to \[- 1\ \] ?

Answer 1

Hint: In this question, we need to explain why both the values of \[\cos(\pi)\] and \[\cos( - \pi)\] are equal to \[- 1\] . Mathematically, \[Pi\ (\pi)\] is Greek letter and is a mathematical constant . In trigonometry, the value of \[\pi\] is \[180^{o} \]. Cosine is nothing but a ratio of the adjacent side of a right angle to the hypotenuse of the right angle. The basic trigonometric functions are sine , cosine and tangent. First we can find the value of \[\cos(\pi)\] by using trigonometric functions.
Trigonometry table :

Angle	\[0^{o}\]	\[30^{o}\]	\[45^{o}\]	\[60^{o}\]	\[90^{o}\]
Cosine	\[1\]	\[\dfrac{\sqrt{3}}{2}\]	\[\dfrac{1}{\sqrt{2}}\]	\[\dfrac{1}{2}\]	\[0\]

Identity used :
1.\[\cos(180^{o} – x)\ = - \cos x\]
2.\[\cos( - \ \theta) = cos\ \theta\]

Complete step by step solution:
Given, \[\cos(pi)\]
We know that the value of \[\pi\] is \[180^{o}\] , by substituting the value of \[\pi\] in \[\cos\left({pi} \right)\]
We get, \[\cos(pi)\ = \ cos(180^{o})\]
Now we can find the value of \[\cos(180^{o})\] by using other angles of cosine functions.
We can rewrite \[180^{o}\] as \[(180^{o} – 0^{o})\] ,
\[\cos(180^{o})\ = cos(180^{o} – 0^{o})\]
From the trigonometric identity, We know that
\[\cos(180^{o} – x)\ = - \ cos\ x\]
We get ,
\[\cos(180^{o} – 0^{o})\ = - cos(0^{o})\]
From the trigonometry table, we know that the value of \[\cos 0^{o}\] is \[1\] .
Thus we get,
\[- cos(0^{o})\ = - 1\]
Therefore we get the value of \[\cos(180^{o})\] is \[- 1\] .
Now we can find the value of \[\cos( - pi)\],
The value of \[\pi\] is \[180^{o} \], by substituting the value we get \[\cos( - pi)\ = cos( - 180^{o})\]
From the trigonometric identity we know that \[\cos( - \ \theta) = cos\ \theta\]
Thus we get \[\cos( - 180^{o})\ = cos\ 180^{o}\]
We have already found the value of \[\cos\ 180^{o}\] is equal to \[- 1\] .
From this we can conclude that the value of \[\cos(pi)\] and \[\cos( - pi)\] both equal to \[- 1\] .
One of the reasons is because cosine is an even function. That means \[\cos( - \theta)\ = cos\ \theta\ \]
Thus the value of \[\cos(pi)\] and \[\cos( - pi)\] both equal to \[- 1\].
Final answer :
Since cosine is an even function the value of \[\cos(pi)\] and \[\cos( - pi)\] both equal to \[- 1\].

Note: The concept used in this problem is trigonometric identities and ratios. Trigonometric identities are nothing but they involve trigonometric functions including variables and constants. The common technique used in this problem is the use of trigonometric functions. We know that a full circle is \[2\pi\] where \[\pi\] is halfway around the circle in clockwise direction and \[- \pi\] in anti clockwise direction.