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Question: How do you evaluate \(\cos \left( -\dfrac{\pi }{3} \right)\)?...

How do you evaluate cos⁡(−π3)\cos \left( -\dfrac{\pi }{3} \right)?

Explanation

Solution

If we want to solve the trigonometric identity cos⁡(−π3)\cos \left( -\dfrac{\pi }{3} \right)then remember the table of the angles and their identities. Put the value of the identities of their angles. Use the formula for the cosine that iscos⁡(−θ)=cos⁡θ\cos \left( -\theta \right)=\cos \theta . Here the negative angle is converted into positive because the negative angle implies that the angle is in the fourth quadrant. In the fourth quadrant any angle of cosine whether it is positive or negative, the result is always positive. So cos⁡(−θ)=cos⁡(2π−θ)=cos⁡θ\cos \left( -\theta \right)=\cos \left( 2\pi -\theta \right)=\cos \theta .

Complete step by step solution:
We have our given identity that is cos⁡(−π3).....(1)\cos \left( -\dfrac{\pi }{3} \right).....\left( 1 \right).
We have to use the formula cos⁡(−θ)=cos⁡θ\cos \left( -\theta \right)=\cos \theta , we have to write the identity into the simplified form such that, we get:
⇒cos⁡(−π3) ⇒cos⁡π3.....(2) \begin{aligned} & \Rightarrow \cos \left( -\dfrac{\pi }{3} \right) \\\ & \Rightarrow \cos \dfrac{\pi }{3}.....\left( 2 \right) \\\ \end{aligned}
Now, we have the identity (2). The identities are converted in simplified form because it will be easier to solve.
⇒cos⁡(−π3) ⇒cos⁡π3.....(3) \begin{aligned} & \Rightarrow \cos \left( -\dfrac{\pi }{3} \right) \\\ & \Rightarrow \cos \dfrac{\pi }{3}.....\left( 3 \right) \\\ \end{aligned}
Now, we have obtained the identity (3). The angleπ3\dfrac{\pi }{3}is 60∘60{}^\circ in degrees form. We know that cos⁡π3\cos \dfrac{\pi }{3}is equal to12\dfrac{1}{2}. So apply it in the identity.
⇒cos⁡π3 ⇒12.....(4) \begin{aligned} & \Rightarrow \cos \dfrac{\pi }{3} \\\ & \Rightarrow \dfrac{1}{2}.....\left( 4 \right) \\\ \end{aligned}

Now, we have obtained the solution to the problem in identity (4). The solution of the given identity is 12\dfrac{1}{2}.

Note: While solving trigonometric identities we should keep in mind that there are 4 quadrants. In the first quadrant, all the trigonometric identities are positive, in the second quadrant all identities are negative except sine, in the third all are negative except tan and in the fourth quadrant all are negative except cosine.