Trigonometry and Geometry Concepts (Flashcards)

Similarity, Ratios, and Triangles

Ratios measure relationships; not changed by scaling. Use to compare A to B via A:B.
Similar triangles: same angles, proportional sides; size can differ.
Scaling example: if top width = 6 and overall height = 10, then height-to-width ratio equals 10:6 = 5:3, so Y = (5/3)X.
Triangles are rigid (don’t deform under motion); many shapes (e.g., squares, pentagons) can bend, but triangles remain similar under scaling.
Right triangles: define a reference angle θ. Sides: opposite, adjacent, hypotenuse.
Trig ratios:
- \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}
- \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}
- \tan \theta = \frac{\text{opposite}}{\text{adjacent}}
Reciprocal relations:
- \tan \theta = \frac{\sin \theta}{\cos \theta}
- \sec \theta = \frac{1}{\cos \theta}
Why the name "co" in cosine: complementary angle to sine.
Core identities (from the Pythagorean theorem):
- \sin^2 \theta + \cos^2 \theta = 1 (Pythagorean identity)
- \tan^2 \theta + 1 = \sec^2 \theta
Double-angle identities:
- \sin 2x = 2\sin x\cos x
- \cos 2x = \cos^2 x - \sin^2 x
- Alternative forms: \cos 2x = 2\cos^2 x - 1 = 1 - 2\sin^2 x
Power reduction (sine/cosine powers):
- \sin^2 x = \frac{1 - \cos 2x}{2}, \quad \cos^2 x = \frac{1 + \cos 2x}{2}
These identities let you rewrite expressions in multiple ways to simplify problems.

Two angle measures:
- Degrees: 360° = 1 full revolution.
- Radians: 180° = (\pi) radians; 2(\pi) radians = 360°.
Conversion: \theta\text{(radians)} = \theta\text{(degrees)} \cdot \frac{\pi}{180}
Circumference and pi: circumference = 2\pi R; pi defined as circumference/diameter.
Unit: radians are often treated as unitless in calculus; convenient for limits and infinitesimals.

Key angles and trig values:
- 0: \sin 0 = 0, \ \cos 0 = 1, \ \tan 0 = 0
- $\dfrac{\pi}{6}$ (30°): \sin = \tfrac{1}{2}, \ \cos = \tfrac{\sqrt{3}}{2}, \ \tan = \tfrac{\sqrt{3}}{3}
- $\dfrac{\pi}{4}$ (45°): \sin = \tfrac{\sqrt{2}}{2}, \ \cos = \tfrac{\sqrt{2}}{2}, \ \tan = 1
- $\dfrac{\pi}{3}$ (60°): \sin = \tfrac{\sqrt{3}}{2}, \ \cos = \tfrac{1}{2}, \ \tan = \sqrt{3}
- $\dfrac{\pi}{2}$ (90°): \sin = 1, \ \cos = 0, \ \tan \text{ undefined (infinite)}
Additional: \sin \pi = 0, \ \cos \pi = -1, \ \sin \tfrac{3\pi}{2} = -1, \ \cos \tfrac{3\pi}{2} = 0.

On a unit circle (radius = 1): point at angle $\theta$ has coordinates $(\cos\theta, \sin\theta)$.
Examples: at $\theta = \pi$, coordinates = (-1, 0); at $\theta = \tfrac{3\pi}{2}$, coordinates = (0, -1).
Relationships:
- X-coordinate = \cos \theta, Y-coordinate = \sin \theta
- Sine is an odd function; cosine is an even function; tangent is odd.

Sine: periodic with period 2\pi; bounded between -1 and 1; odd function (sine(-θ) = -sine(θ)).
Cosine: period 2\pi; even function (cos(-θ) = cos(θ)); starts at 1 when θ = 0.
Tangent: period \pi; odd function; has vertical asymptotes at \tfrac{\pi}{2} + k\pi.
Inverse tangent (arctan): defined for all real inputs; tan is not one-to-one unless restricted to a branch.
- Notation caveat: tan^{-1} is arctan, not 1/tan.
- Examples: arctan(0) = 0; arctan(1) = (\pi/4); arctan(∞) → (\pi/2).

Sum formulas:
- \sin(X+Y) = \sin X\cos Y + \cos X\sin Y
- \cos(X+Y) = \cos X\cos Y - \sin X\sin Y
Special case X = Y:
- \sin 2X = 2\sin X\cos X
- \cos 2X = \cos^2 X - \sin^2 X
Alternative double-angle forms:
- \cos 2X = 2\cos^2 X - 1 = 1 - 2\sin^2 X
Power reduction forms:
- \sin^2 X = \frac{1 - \cos 2X}{2}, \quad \cos^2 X = \frac{1 + \cos 2X}{2}
These allow rewriting expressions to simplify calculations.

Distance between points: d = \sqrt{(x1 - x0)^2 + (y1 - y0)^2}
Circle equation: (x-h)^2 + (y-k)^2 = R^2 (points at distance R from center (h,k))
Circle basics:
- Circumference: 2\pi R
- Area: \pi R^2
Other shapes:
- Cylinder volume: V = \pi r^2 h
- Sphere volume: V = \tfrac{4}{3}\pi r^3

Classic area proof: four copies of a right triangle form a large square; inner square of side C; area equation leads to A^2 + B^2 = C^2.
Scaling idea: scaling by factor X scales lengths by X and areas by X^2; this underpins similar-triangle arguments.
Alternate similarity-based proof outline: two smaller triangles similar to the original lead to the same relation, yielding a playful connection to the phrase "MC squared".
Angle-sum intuition (brief): constructing adjacent right triangles helps derive the angle-sum formulas for sine and cosine; reinforces why \sin(X+Y) = \sin X\cos Y + \cos X\sin Y and \cos(X+Y) = \cos X\cos Y - \sin X\sin Y hold.

Trig is built on ratios that are preserved under scaling; use similar triangles to relate lengths.
Sine and cosine encode coordinates on the unit circle: X = \cos \theta, \ Y = \sin \theta.
Sine and cosine are periodic with period 2\pi; tangent has period \pi and has vertical asymptotes.
Key identities: \sin^2 \theta + \cos^2 \theta = 1, \tan^2 \theta + 1 = \sec^2 \theta, double-angle and power-reduction formulas as listed above.
Special-angle values at 0, \dfrac{\pi}{6}, \dfrac{\pi}{4}, \dfrac{\pi}{3}, and \dfrac{\pi}{2} provide quick reference for common calculations.
Basic distance and circle formulas: distance, circle equation, circumference, and area relationships.
Pythagorean theorem is foundational and can be visualized through area or similarity arguments; it also connects to proofs of angle-sum identities.