Question

How do you find the exact value of the trigonometric function $\cos {570}^{\circ }$ ?

Answer 1

Hint: In this question, we have to find the value of the cosine of 570 degrees, cosine is a trigonometric function. So to solve the question, we must know the details of trigonometric functions and how to find their values. The trigonometry helps us to find the relation between the sides of a right-angled triangle that is the base, the perpendicular and the hypotenuse. The trigonometric functions repeat a pattern after 360 degrees, as the given angle is greater than 360 degrees so we subtract 360 degrees from the given angle, and then find the cosine of the angle obtained.

Complete step by step answer:
Let us find the correct answer,
$$\begin{gathered} \Rightarrow \cos ({570}^{\circ })=\cos ({360}^{\circ }+{210}^{\circ }) \\ \Rightarrow \cos ({570}^{\circ })=\cos ({210}^{\circ }) \end{gathered}$$
Now,
$\Rightarrow \cos ({210}^{\circ })=\cos ({180}^{\circ }+{30}^{\circ })$
We know that cosecant is negative in the third quadrant, so –
$$\begin{gathered} \Rightarrow \cos ({570}^{\circ })=-\cos ({30}^{\circ }) \\ \Rightarrow \cos ({570}^{\circ })=-{\displaystyle \frac{\sqrt{3}}{2}} \end{gathered}$$
Hence, the exact functional value of $\cos ({570}^{\circ })$ is $-{\displaystyle \frac{\sqrt{3}}{2}}$ .

Note: All the trigonometric functions have a positive value in the first quadrant. In the second quadrant, sine is positive while all the other functions are negative, in the third quadrant, tan function is positive while all other functions are positive and in the fourth quadrant, cosine function is positive while all others are negative that’s why $\cos (180+\theta )=-\cos \theta$ and we also know that the trigonometric functions are periodic. We know the value of the cosine function when the angle lies between 0 and ${\displaystyle \frac{\pi }{2}}$ . That’s why we use the periodic property of these functions to find the value of the cosine of the angles greater than ${\displaystyle \frac{\pi }{2}}$ .