2d
Spending 10 hours on mathematics and arriving at 6 trigonometric identity derivations might feel like slow progress, especially if you were aiming for more. But let's dissect this with care and realism.
1. Depth Over Quantity
If you truly derived 6 trigonometric identities from scratch — not just memorized them — then you've done something meaningful. Understanding derivations helps:
Cement intuition for how sine, cosine, etc. interrelate.
Build a mental toolkit for more advanced math (e.g., Fourier analysis, complex numbers, physics).
Improve symbolic manipulation and algebraic maturity.
Mathematics isn’t a race — it’s often about slow, deep burns rather than fast results.
Analogy: Think of it like metallurgy — forging a few strong blades is better than mass-producing brittle ones.
2. What Counts as ‘Only’?
There’s a subtle but powerful shift when you stop measuring success by quantity and start asking:
“Did I understand something I didn’t before?”
“Can I reproduce or explain it without notes?”
“Did I uncover how things are connected?”
If you came away with insight, even if it's just one deep insight, that’s success.
3. Cognitive Load in Trig
Trig identities are deceptive. They're short, but often hide layers of insight. If you tackled things like:
sin
2
𝑥
+
cos
2
𝑥
=
1
sin
2
x+cos
2
x=1
tan
𝑥
=
sin
𝑥
cos
𝑥
tanx=
cosx
sinx
Angle addition formulas:
sin
(
𝑎
+
𝑏
)
=
sin
𝑎
cos
𝑏
+
cos
𝑎
sin
𝑏
sin(a+b)=sinacosb+cosasinb
...you might have wrestled with:
Geometric intuition (unit circle, triangles)
Algebraic manipulation
Coordinate transformations
Symbol juggling under pressure
That’s not trivial work. If you derived even the angle addition formulas from geometry or from Euler’s formula, that’s deep.
4. Mental Fatigue Is Real
Mathematics uses intense working memory and symbolic manipulation. Ten hours of that is a cognitive marathon. You’re not just solving problems — you’re training an internal symbolic engine. That can feel “unproductive” because insight is often nonlinear:
You grind 6 hours and nothing clicks. Then suddenly in hour 9, it crystallizes — and you're never the same again.
5. Build a Narrative
If this was a journal entry, it might read like this:
“Day X: Derived 6 trig identities. Took longer than I expected. But I now understand how cosine difference emerges from the geometry of the unit circle, and how sine and cosine reflect orthogonal projections. The angle addition formula isn't magic — it's a rotated triangle in disguise.”
That’s valuable narrative data. You're not just learning trig — you're watching yourself evolve.
TL;DR Comments:
Don’t discount small visible outputs — they often mask huge invisible internal upgrades.
6 true derivations = 6 moments of clarity = serious work.
Fatigue is part of the terrain. You’re conditioning your mind.
Build from here: Next time, these identities will be tools — not puzzles.
Would you like help creating a visualization of how these identities interconnect? Or would you like to share one derivation you’re proud of and we analyze it?
1. Depth Over Quantity
If you truly derived 6 trigonometric identities from scratch — not just memorized them — then you've done something meaningful. Understanding derivations helps:
Cement intuition for how sine, cosine, etc. interrelate.
Build a mental toolkit for more advanced math (e.g., Fourier analysis, complex numbers, physics).
Improve symbolic manipulation and algebraic maturity.
Mathematics isn’t a race — it’s often about slow, deep burns rather than fast results.
Analogy: Think of it like metallurgy — forging a few strong blades is better than mass-producing brittle ones.
2. What Counts as ‘Only’?
There’s a subtle but powerful shift when you stop measuring success by quantity and start asking:
“Did I understand something I didn’t before?”
“Can I reproduce or explain it without notes?”
“Did I uncover how things are connected?”
If you came away with insight, even if it's just one deep insight, that’s success.
3. Cognitive Load in Trig
Trig identities are deceptive. They're short, but often hide layers of insight. If you tackled things like:
sin
2
𝑥
+
cos
2
𝑥
=
1
sin
2
x+cos
2
x=1
tan
𝑥
=
sin
𝑥
cos
𝑥
tanx=
cosx
sinx
Angle addition formulas:
sin
(
𝑎
+
𝑏
)
=
sin
𝑎
cos
𝑏
+
cos
𝑎
sin
𝑏
sin(a+b)=sinacosb+cosasinb
...you might have wrestled with:
Geometric intuition (unit circle, triangles)
Algebraic manipulation
Coordinate transformations
Symbol juggling under pressure
That’s not trivial work. If you derived even the angle addition formulas from geometry or from Euler’s formula, that’s deep.
4. Mental Fatigue Is Real
Mathematics uses intense working memory and symbolic manipulation. Ten hours of that is a cognitive marathon. You’re not just solving problems — you’re training an internal symbolic engine. That can feel “unproductive” because insight is often nonlinear:
You grind 6 hours and nothing clicks. Then suddenly in hour 9, it crystallizes — and you're never the same again.
5. Build a Narrative
If this was a journal entry, it might read like this:
“Day X: Derived 6 trig identities. Took longer than I expected. But I now understand how cosine difference emerges from the geometry of the unit circle, and how sine and cosine reflect orthogonal projections. The angle addition formula isn't magic — it's a rotated triangle in disguise.”
That’s valuable narrative data. You're not just learning trig — you're watching yourself evolve.
TL;DR Comments:
Don’t discount small visible outputs — they often mask huge invisible internal upgrades.
6 true derivations = 6 moments of clarity = serious work.
Fatigue is part of the terrain. You’re conditioning your mind.
Build from here: Next time, these identities will be tools — not puzzles.
Would you like help creating a visualization of how these identities interconnect? Or would you like to share one derivation you’re proud of and we analyze it?
