Trigonometric Functions (Unit Circle Definition): Let \(t\) be a real number representing the arc length from the point (1, 0) to a point \(P(x, y)\) on the unit circle. The trigonometric functions of \(t\), also called circular functions, are defined by the coordinates of \(P\). For example, \(\cos(t) = x\) and \(\sin(t) = y\).
Theorems
Lemma: Let \(\theta\) be an angle in standard position, and suppose the terminal side of \(\theta\) intersects the unit circle at the point \(P(x, y)\). Then \(\cos(\theta) = x\) and \(\sin(\theta) = y\).
Domains of the Circular Functions: Let \(t\) be a real number and \(k\) be an integer. The domains of the circular functions are:
\(\sin(t)\) and \(\cos(t)\): All real numbers, or \((-\infty, \infty)\)
\(\tan(t)\) and \(\sec(t)\): All real numbers except \(t = \frac{\pi}{2} + k\pi\)
\(\cot(t)\) and \(\csc(t)\): All real numbers except \(t = k\pi\)
Ranges of the Circular Functions: Let \(t\) be a real number. The ranges of the circular functions are:
\(\sin(t)\) and \(\cos(t)\): \([-1, 1]\)
\(\tan(t)\) and \(\cot(t)\): \((-\infty, \infty)\)
\(\sec(t)\) and \(\csc(t)\): \((-\infty, -1] \cup [1, \infty)\)
Common Mistakes
Using the Wrong Calculator Mode: When working with the unit circle definition, angles and arc lengths are measured in radians. You must ensure your calculator is set to Radian mode, not Degree mode, to get correct approximations.
Confusing Angle Measure with Arc Length: While the numerical value of an angle in radians is equal to the arc length it subtends on a unit circle, it is important to remember they are conceptually different. An angle is an amount of rotation, while an arc length is a distance.