“Unit Circle” Trigonometry

the unit circle in the first quadrant, with an arc of length t and the corresponding terminal point labelled as (cos(t),sin(t)) An arc of a circle is a segment of its circumference, and the length of an arc is referred to as its arclength. The “slice” of circle subtended by that arc is called a sector of the circle. Given a real number \(t\) the terminal point corresponding to \(t\) is the point on the unit circle you get by starting at \((1,0)\) (the rightmost side) and moving counterclockwise by an arclength of \(t.\) Negative values of \(t\) correspond to clockwise movement. The area of the sector with terminal point \(t\) will be \(\frac{1}{2}t.\) the unit circle in the first quadrant with the values of sine and cosine labelled for all multiples of pi/6 and pi/4 The reference number \(\overline{t}\) of a terminal point \(t\) is the number between \(0\) and \(2\pi\) that has the same terminal point as \(t.\) The trigonometric functions sine \((\sin)\) and cosine \((\cos)\) return the coordinates of a terminal point: \(\cos(t)\) returns the \(x\)-coordinates of the terminal point of \(t\) and \(\sin(t)\) returns the \(y\)-coordinates of the terminal point of \(t.\) The values of \(\sin(t)\) and \(\cos(t)\) must be between \(-1\) and \(1.\) Note that \(\cos(-t) = \cos(t)\) and \(\sin(-t) = -\sin(t).\) The values of sine and cosine at values of \(t\) that are a multiple of \(\frac{\pi}{6}\) or \(\frac{\pi}{4}\) have simple algebraic presentations worth memorizing. There are four other trigonometric functions to be aware of, tangent and secant and cosecant and cotangent, as well as three Pythagorean identities relating these trigonometric functions.

\(\displaystyle \tan(t) \!=\! \frac{\sin(t)}{\cos(t)}\)
\(\displaystyle \sec(t) \!=\! \frac{1}{\cos(t)}\)
\(\displaystyle \csc(t) \!=\! \frac{1}{\sin(t)}\)
\(\displaystyle \cot(t) \!=\! \frac{1}{\tan(t)}\)
\(\displaystyle \sin^2(t) \!+\! \cos^2(t) \!=\! 1\)
\(\displaystyle \tan^2(t) \!+\! 1 \!=\! \sec^2(t)\)
\(\displaystyle 1 \!+\! \cot^2(t) \!=\! \csc^2(t)\)

The tangent function could be better thought of as returning the slope of the line through the terminal point of \(t.\) Typographic notes: Some authors write their trig functions without parenthesis: \(\sin t\) and \(\sin(t)\) mean the same thing. Also it’s conventional to write \(\sin^2(t)\) for \(\bigl(\sin(t)\bigr)^2.\) However \(\sin^{-1}(t) \neq \bigl(\sin(t)\bigr)^{-1};\) a power of \(-1\) on a function denotes its inverse function.