Trigonometry by Prof Rob Bob

About this course

Trigonometry feels much easier once you can see how angles, triangles, and circles tell the same story. This free online course helps you build that connected understanding, step by step, so you can move from memorizing to actually reasoning. You will start by making sense of standard position angles, coterminal angles, and the everyday conversions between degrees and radians, including degree-minute-second notation. With those fundamentals in place, you will be able to interpret angle measure fluently and avoid the common mistakes that slow students down.

From there, the unit circle becomes a powerful tool rather than a diagram to recite. You will learn how reference angles and quadrants control signs, and how sine and cosine correspond to coordinates on the circle. This leads naturally into evaluating trig functions accurately, connecting angular speed and linear speed, and applying arc length ideas to real scenarios. As your confidence grows, you will practice rewriting and simplifying expressions using reciprocal, quotient, and Pythagorean identities, including strategies for evaluating expressions without relying on a calculator.

The course also strengthens your right-triangle skills: solving for missing sides, finding acute angles with inverse trig, and using cofunctions and complementary relationships correctly. Then you will expand to trigonometric functions of any angle, where quadrant reasoning becomes essential, and you will develop intuition for graphs. By working with transformations, periods, and key features of sine, cosine, tangent, and secant, you will learn to read and build equations from graphs and to model real situations with periodic functions.

As you progress, you will practice proving and verifying identities, using sum and difference formulas, half-angle and double-angle relationships, and product-to-sum techniques. You will also solve trigonometric equations with both restricted intervals and general solutions, learning how to capture every valid angle. Finally, the course broadens into applications that often appear in advanced classes: oblique triangles with the Law of Sines and Law of Cosines, area formulas including Heron’s method, polar coordinates and conversions, complex numbers in polar form with De Moivre’s Theorem, and vectors with dot products, projections, and navigation-style bearing problems. By the end, you will have a complete, practical trigonometry toolkit built through explanation, repetition, and targeted exercises.