The Unshakable Identity
Why sin²x + cos²x is always 1, no matter how x changes.
Equation
y = sin(x)^2 + 0*cos(x)^2
Graph
Table
| x | y |
|---|---|
| -3.14 | 0 |
| -2.64 | 0.23 |
| -2.14 | 0.71 |
| -1.64 | 0.99 |
| -1.14 | 0.83 |
| -0.64 | 0.36 |
| -0.14 | 0.02 |
| 0.36 | 0.12 |
| 0.86 | 0.57 |
| 1.36 | 0.96 |
| 1.86 | 0.92 |
| 2.36 | 0.5 |
| 2.86 | 0.08 |
What this lesson covers
What you do
You shape the function y = sin(x)^2 + k*cos(x)^2 and watch the graph answer.
Challenges to clear
- Slide k: ONLY k = 1 collapses the wave into the flat line y = 1 — sin²x + cos²x = 1, always. Check x = 0.5.
- Still exactly 1 at x = 2 — the identity is unshakable at every angle.
- Scale the sine part: slide a to 3.
- Now slide k until the wave flattens completely onto y = 3. Only equal weights work — 3sin²x + 3cos²x = 3 at every angle.
Check yourself
Why does the graph of y = sin²x + cos²x look like a flat line at y=1?
- Because sin and cos are always equal to each other.
- Because the sum of their squares is constant for any angle. — correct
- Because the calculator rounds the answer to 1.
- Because sin²x and cos²x are always positive numbers.
Think about it
- sin²x alone waves up and down. What must you ADD to make it flat at 1 for every x?