Learn mathematics
High-school mathematics, one topic at a time: the rules you need, a picture you can move, worked examples step by step, and practice questions that check your answer.
Unit
Trigonometry
Sine, cosine and tangent in a right triangle
What the four ratios mean, the two triangles that give every special angle, and the square-root trick for remembering the table.
InteractivePractice2TrigonometryThe unit circle: sin, cos, tan and cot as lengths
Turn the angle and see where each value lives on the circle, which quadrant makes it positive, and why tan and cot are sometimes undefined.
InteractivePractice3TrigonometryAngles beyond 90°: reference angles and signs
How 150° borrows its value from 30°, which functions are positive in each quadrant, and the identities for 180° − a, 180° + a and 360° − a.
InteractivePractice4TrigonometryTrigonometric identities and simplifying
The Pythagorean identity and its two rearrangements, tan and cot as quotients, the square identities, and how to simplify a fraction of sines and cosines.
PracticeUnit
Triangles
Pythagoras and the triples worth knowing
Why the squares on the legs fill the square on the hypotenuse, the triples 3-4-5, 5-12-13, 8-15-17 and 7-24-25, and the k-method for scaled triangles.
InteractivePractice6TrianglesThe angle bisector theorem
An angle bisector cuts the opposite side in the ratio of the other two sides — inside and outside the triangle — and every point on it is equidistant from the two arms.
InteractivePractice7TrianglesMedians, the centroid and the midsegment
The three medians meet at the centroid G, which cuts each median 2 : 1 from the vertex; the segment joining two midpoints is parallel to the third side and half as long.
InteractivePractice8TrianglesArea of a triangle: ratios and the sine formula
Triangles with the same height have areas in the ratio of their bases; with two sides and the angle between them, Area = ½ · a · b · sin C.
InteractivePractice9TrianglesThe sine rule and the cosine rule
a / sin A = b / sin B = c / sin C, and a² = b² + c² − 2bc · cos A: which one to use, how to solve any triangle, and a solver that draws the result.
InteractivePracticeWhat these lessons cover
The trigonometry unit builds from the right triangle to the unit circle, then to angles beyond 90° and the identities used for simplifying. The triangles unit covers Pythagoras and the triples, the angle bisector theorem, medians and the centroid, area ratios and the sine area formula, and the sine and cosine rules. Each lesson has a figure you can move and practice questions with instant checking; the scientific calculator and the function plotter are a click away when you want to check a number.