Precalculus
Every lesson explains the concept, works through examples, and gives you practice problems with full solutions. Work through them in order, or jump to the topic you need.
17 lessons
- Verifying Trigonometric IdentitiesProve that two trigonometric expressions agree for every angle: work one side at a time, rewrite in sines and cosines, and trade with the Pythagorean identities.
- Sum and Difference Formulas for Sine, Cosine and TangentWhy cos(α − β) = cos α cos β + sin α sin β, proved by rotating a chord of the unit circle, how the other formulas follow, and how to find exact values like cos 15°.
- Double-Angle and Half-Angle FormulasDerive sin 2θ, cos 2θ and tan 2θ from the sum formulas, reverse them into half-angle and power-reducing forms, and use them for exact values and equations.
- Inverse Trigonometric FunctionsWhy sine, cosine and tangent need a restricted domain before they can be undone, what arcsin, arccos and arctan return, and how to solve trig equations in context.
- Vectors: Magnitude, Direction and ComponentsWhat makes a quantity a vector, how to draw and name one, why its components come from subtracting the start from the end, and how to use vectors for velocity.
- Adding, Subtracting and Scaling VectorsHow to add vectors end to end and by components, why the length of a sum is usually less than the sum of the lengths, how subtraction works, and what scaling does.
- The Dot Product and the Angle Between VectorsMultiply two vectors into a number: u · v = u₁v₁ + u₂v₂ equals |u||v|cos θ, which measures the angle between them, tests for right angles and gives projections.
- Matrices: Operations, Identity and InversesHow matrices store data, how to add, scale and multiply them, why the order of multiplication matters, and how the determinant decides whether an inverse exists.
- Matrices as Transformations of the PlaneHow a 2×2 matrix moves every vector in the plane, why its columns show where the whole plane goes, and why the determinant measures how it scales area.
- Solving Systems with Matrix EquationsHow to write a system of linear equations as a single matrix equation AX = B, why multiplying by the inverse solves it, and what a zero determinant says about the system.
- Ellipses and Hyperbolas from Their FociWhy an ellipse is every point whose distances to two foci add to a constant, how squaring twice turns that into x²/a² + y²/b² = 1, and how the hyperbola follows.
- Conic Sections in General FormTell a circle, ellipse, parabola and hyperbola apart from Ax² + Cy² + Dx + Ey + F = 0, then complete the square to find the center, radius, vertices and axes.
- Parametric EquationsDescribe a curve by giving x and y as functions of a parameter t: plotting and direction, eliminating the parameter, line segments and projectile motion.
- Polar Coordinates and Polar GraphsLocate points by distance and angle, convert between polar and rectangular coordinates, and graph polar equations: circles, cardioids and roses.
- The Complex Plane: Polar Form and Geometric OperationsHow to plot complex numbers, why rectangular and polar form name the same point, why multiplying multiplies lengths and adds angles, and how to find distances.
- De Moivre's Theorem and Roots of Complex NumbersRaise a complex number to a power by raising its length and multiplying its angle, then reverse the rule to find all n of its nth roots, evenly spaced on a circle.
- Proof by Mathematical InductionProve a statement for every positive integer with two steps: check it for n = 1, then show that whenever it holds for k it holds for k + 1.