Chapter 5: Trigonometric Functions: Unit Circle Approach
Chapter 5 builds trig from scratch using the unit circle — a circle with radius 1 centered at the origin. Instead of triangles, we use a point moving around that circle to define sine, cosine, and their relatives. This is the foundation everything else in trig is built on.
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Sections
5.1The Unit Circle
The unit circle is a circle of radius 1 centered at the origin. Every point on it has coordinates (cos t, sin t) where t is the angle. Memorizing the key points on this circle — at 30°, 45°, 60° intervals — is the foundation of all of trig.
5.2Trigonometric Functions of Real Numbers
Now we give names to the coordinates of terminal points. Sine is the y-coordinate, cosine is the x-coordinate. From those two, we get four more: tangent, cotangent, secant, and cosecant. All six trig functions come from one point on the unit circle.
5.3Trigonometric Graphs
When you graph sine and cosine, you get waves. This section is about understanding and transforming those waves — stretching them taller (amplitude), making them repeat faster or slower (period), and sliding them left or right (phase shift). These transformations show up everywhere in physics and engineering.
5.4More Trigonometric Graphs
Tangent, cotangent, secant, and cosecant have graphs too — but they look very different from sine and cosine. Instead of smooth waves, they have vertical asymptotes (places where the function blows up to ±∞) and the graphs come in separate pieces.
5.5Inverse Trigonometric Functions
The regular trig functions take an angle and give you a ratio. Inverse trig functions do the reverse: you give them a ratio and they give you the angle. But because trig functions repeat, we have to restrict their domains to make the inverse work.
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