Trigonometric Functions

Trigonometric functions describe periodic phenomena. In Function Guessr, they appear as repeating waves.

$f(x) = A \sin(B(x - C)) + D$

The Big Three

1. Sine ($\sin x$)

• Starts at: $(0,0)$ (for standard sine).
• Shape: Smooth wave.
• Range: $[-1, 1]$.

2. Cosine ($\cos x$)

• Starts at: $(0,1)$ (for standard cosine).
• Shape: Same shape as sine, just shifted.
• Identity: $\cos(x) = \sin(x + \pi/2)$.

3. Tangent ($\tan x$)

• Shape: Repeating "S" curves separated by vertical asymptotes.
• Asymptotes: At $x = \pi/2 + n\pi$.
• Range: $(-\infty, \infty)$.

🧠 Advanced Guessing Strategies

Strategy 1: Find the Amplitude ($A$)

The amplitude is half the distance between the peak and the trough.

1. Look for the maximum value ($y_{max}$) and minimum value ($y_{min}$).
2. $A = \frac{y_{max} - y_{min}}{2}$
3. If the wave goes from $-3$ to $3$, the amplitude is $3$. You likely have $3\sin(x)$ or $3\cos(x)$.

Strategy 2: Determine the Period ($P$)

The period is the length of one full cycle.

1. Find two consecutive peaks (or x-intercepts).
2. Measure the distance between them.
3. The coefficient $B$ inside the function $f(Bx)$ is related to the period $P$ by: * For $\sin, \cos$: $P = \frac{2\pi}{|B|}$ * For $\tan$: $P = \frac{\pi}{|B|}$

Tip: If the wave repeats every $\pi$ (approx 3.14) units instead of $2\pi$ (approx 6.28), then $B=2$. The function is $\sin(2x)$.

Strategy 3: Phase Shift vs. Identity

Is it $\sin(x)$ shifted or just $\cos(x)$?

• Check $x=0$.
  • $f(0) = 0 \rightarrow$ Likely Sine.
  • $f(0) = 1 \rightarrow$ Likely Cosine.
• Function Guessr prefers simple forms. If it looks like a cosine, guess $\cos(x)$ before trying $\sin(x + \pi/2)$.

Strategy 4: Tangent Asymptotes

If the function shoots off to $\pm \infty$ repeatedly, it's likely $\tan(x)$.

• Check where the vertical lines (asymptotes) are.
• Standard $\tan(x)$ explodes at $\pi/2 \approx 1.57$.