Fundamentals
The core facts every aspirant should own — each a titled nugget with a real-world story, the concept in plain words, and a memory trick. Works even when the internet doesn't.
1,200 fundamentals
“GPS pinpoints you by solving triangles between satellites — trigonometry from space.”
A frequent error is using degrees in derivative/limit formulas. In reality, angles can be measured in degrees or radians (π rad = 180°); calculus formulas assume radians.
Memory trick: always switch to radians before differentiating trig functions.
“GPS pinpoints you by solving triangles between satellites — trigonometry from space.”
A frequent error is accepting an answer where sinθ > 1. In reality, sinθ and cosθ always lie between −1 and 1.
Memory trick: if a step gives sinθ = 1.4, you've made an error somewhere.
“GPS pinpoints you by solving triangles between satellites — trigonometry from space.”
The fundamental Pythagorean identity. Use it when any real angle θ.
sin²θ + cos²θ = 1
“GPS pinpoints you by solving triangles between satellites — trigonometry from space.”
The sine addition formula. Use it when compound angles.
sin(A±B) = sinA cosB ± cosA sinB
“GPS pinpoints you by solving triangles between satellites — trigonometry from space.”
Degrees split a circle into 360 parts; radians measure the arc length per unit radius (2π in a full circle) and are used in calculus.
“GPS pinpoints you by solving triangles between satellites — trigonometry from space.”
The sine rule (a/sinA = b/sinB) suits two angles and a side; the cosine rule (c²=a²+b²−2ab·cosC) suits two sides and the included angle.
“GPS pinpoints you by solving triangles between satellites — trigonometry from space.”
The range of both sine and cosine is [−1, 1] — a value outside means a mistake.
“Every 'rate' you plot — speed, price, growth — is a straight line's slope telling a story.”
The steepness of a line, m = (y₂−y₁)/(x₂−x₁) = tanθ.
Memory trick: slope is 'rise over run' — change in y over change in x.
“Every 'rate' you plot — speed, price, growth — is a straight line's slope telling a story.”
A frequent error is inverting the formula to (x₂−x₁)/(y₂−y₁). In reality, the steepness of a line, m = (y₂−y₁)/(x₂−x₁) = tanθ.
Memory trick: slope is 'rise over run' — change in y over change in x.
“Every 'rate' you plot — speed, price, growth — is a straight line's slope telling a story.”
Parallel lines have equal slopes; perpendicular lines have slopes whose product is −1.
Memory trick: parallel → equal slopes; perpendicular → m₁·m₂ = −1.
“Every 'rate' you plot — speed, price, growth — is a straight line's slope telling a story.”
A frequent error is thinking perpendicular lines have equal slopes. In reality, parallel lines have equal slopes; perpendicular lines have slopes whose product is −1.
Memory trick: parallel → equal slopes; perpendicular → m₁·m₂ = −1.
“Every 'rate' you plot — speed, price, growth — is a straight line's slope telling a story.”
Slope-intercept y=mx+c, point-slope y−y₁=m(x−x₁), and two-point forms describe the same line.
Memory trick: use point-slope when you have a point and a slope.
“Every 'rate' you plot — speed, price, growth — is a straight line's slope telling a story.”
A frequent error is picking a form that doesn't match the given data. In reality, slope-intercept y=mx+c, point-slope y−y₁=m(x−x₁), and two-point forms describe the same line.
Memory trick: use point-slope when you have a point and a slope.
“Every 'rate' you plot — speed, price, growth — is a straight line's slope telling a story.”
Slope from two points. Use it when x₁ ≠ x₂.
m = (y₂−y₁)/(x₂−x₁)
“Every 'rate' you plot — speed, price, growth — is a straight line's slope telling a story.”
Distance from a point to a line. Use it when line ax+by+c=0.
d = |ax₀+by₀+c|/√(a²+b²)
“Every 'rate' you plot — speed, price, growth — is a straight line's slope telling a story.”
Parallel lines never meet and share the same slope; perpendicular lines cross at 90° with slopes multiplying to −1.
“Every 'rate' you plot — speed, price, growth — is a straight line's slope telling a story.”
Slope-intercept (y=mx+c) is best when you know the slope and y-intercept; point-slope is best when you know the slope and any one point.
“Every 'rate' you plot — speed, price, growth — is a straight line's slope telling a story.”
Equal slopes make lines parallel; perpendicular slopes multiply to −1.
“Satellite dishes are parabolas because a parabola focuses every incoming signal to one point.”
A number e that classifies conics: circle e=0, ellipse e<1, parabola e=1, hyperbola e>1.
Memory trick: as e grows from 0, the shape opens up: circle → ellipse → parabola → hyperbola.
“Satellite dishes are parabolas because a parabola focuses every incoming signal to one point.”
A frequent error is mixing up which conic each eccentricity range gives. In reality, a number e that classifies conics: circle e=0, ellipse e<1, parabola e=1, hyperbola e>1.
Memory trick: as e grows from 0, the shape opens up: circle → ellipse → parabola → hyperbola.
“Satellite dishes are parabolas because a parabola focuses every incoming signal to one point.”
A conic is the set of points whose distances to a focus and a directrix are in the fixed ratio e.
Memory trick: focus is a point inside; directrix is a line outside.
“Satellite dishes are parabolas because a parabola focuses every incoming signal to one point.”
A frequent error is confusing the focus (a point) with the directrix (a line). In reality, a conic is the set of points whose distances to a focus and a directrix are in the fixed ratio e.
Memory trick: focus is a point inside; directrix is a line outside.
“Satellite dishes are parabolas because a parabola focuses every incoming signal to one point.”
Circle (x−h)²+(y−k)²=r² and parabola y²=4ax are the building blocks.
Memory trick: y²=4ax opens right; x²=4ay opens up.
“Satellite dishes are parabolas because a parabola focuses every incoming signal to one point.”
A frequent error is mismatching the axis of the parabola with its equation. In reality, circle (x−h)²+(y−k)²=r² and parabola y²=4ax are the building blocks.
Memory trick: y²=4ax opens right; x²=4ay opens up.