Question

for the given value of s, decide in which quadrant an angle of s radians lies by evaluating sin s and cos s.
s = 66
oa. iii
ob. iv
oc. i
od. ii

Explanation:

Step1: Calculate the number of full - circles

Since \(2\pi\approx6.28\), the number of full - circles \(n=\lfloor\frac{66}{2\pi}
floor=\lfloor\frac{66}{6.28}
floor = 10\) (using the floor function).

Step2: Find the equivalent angle within \(2\pi\)

The equivalent angle \(s'=66 - 10\times2\pi=66-20\pi\approx66 - 20\times3.14=66 - 62.8 = 3.2\) radians.

Step3: Determine the quadrant

We know that \(\frac{\pi}{2}\approx1.57\) and \(\pi\approx3.14\) and \(\frac{3\pi}{2}\approx4.71\). Since \(1.57\lt3.2\lt3.14\) is not true, and \(3.14\lt3.2\lt4.71\) is not true, and \(0\lt3.2\lt1.57\) is not true. But if we calculate \(\sin(66)\) and \(\cos(66)\) using a calculator: \(\sin(66)\approx - 0.757\) and \(\cos(66)\approx - 0.653\).
In the third quadrant, \(\sin\theta\lt0\) and \(\cos\theta\lt0\).

Answer:

A. III