Real Solutions of a Floor-Trig Equation

★ 9/10functionstrigonometryEstimationCasework

Problem

Let f(x)=asin⁡x+bcos⁡xf(x)=a\sin x+b\cos x, where a,b∈Na,b\in\mathbb N, a>2a>2, and

{x∣f(x)=0}={x∣f(f(x))=0}.\{x\mid f(x)=0\}=\{x\mid f(f(x))=0\}.

Find the number of real solutions in (0,30)(0,30) of

(f([x])3)2+(f({x})3)2−1=0,\left(\frac{f([x])}3\right)^2+\left(\frac{f(\{x\})}3\right)^2-1=0,

where [x][x] and {x}\{x\} are the integer and fractional parts of xx.

Answer

Solution

Difficulty9/10
Topicsfunctions, trigonometry, Estimation, Casework

Whiteboard

Your sketch is saved only in this browser. To share it, export your drawing as an image (whiteboard menu → Export as → PNG), then upload that image in the comments below.

Discussion

Ask questions, share alternate solutions, and use LaTeX freely.

0 comments
Log in to join the discussion.

No comments yet.