Question

The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs enclose the same area A. Then A⁴ is equal to __________

Hint:

The area between two consecutive intersections of $\sin x$ and $\cos x$ is always $2\sqrt{2}$ units.

Answer 1

Correct Answer: 64

Step 1: Intersection points: $\sin x = \cos x \Rightarrow x = \pi/4, 5\pi/4, .......$.
Step 2: $A = \int_{\pi/4}^{5\pi/4} (\sin x - \cos x) dx = [-\cos x - \sin x]_{\pi/4}^{5\pi/4}$.
Step 3: $A = [-(-\frac{1}{\sqrt{2}}) - (-\frac{1}{\sqrt{2}})] - [-\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}] = \frac{2}{\sqrt{2}} + \frac{2}{\sqrt{2}} = 2\sqrt{2}$.
Step 4: $A^4 = (2\sqrt{2})^4 = 16 \times 4 = 64$.