Trigonometric Functions - The Complete Academic Support Hub

Unit Circles
Consider the unit circle given by the equation x² + y² = 1. The radius is therefore 1. The equation of the arc length of an angle is s = r where s is the arc length, r is the radius, and ? is the central angle. When the radius is 1, the arc length equals the angle. Using unit circles makes it easier to relate to real life applications.

Trigonometric Functions
Consider the unit circle given by the equation x² + y² = 1. The radius is therefore 1. The equation of the arc length of an angle is s = r where s is the arc length, r is the radius, and is the central angle. When the radius is 1, the arc length equals the angle. Using unit circles makes it easier to relate to real life applications.
Definitions of Trigonometric Functions:
Let t be a real number and let (x, y) be the point on the unit circle corresponding to the angle t. sin t = y cos t = x tan t = y/x x 0 csc t = 1/y y 0 sec t = 1/x x 0 cot t = x/y y 0 Cosecant, secant, and cotangent are the reciprocals of sine, cosine, and tangent respectively. The tangent and secant of t are not defined when x = 0. This is shown in the following examples: t = /2 corresponds to (0, 1) because cos(/2) = 0 and sin(/2) = 1, it follows that tan(/2) and sec(/2) are undefined. The cotangent and cosecant functions are also undefined when y = 0. t = 0 corresponds to (1,0), therefore cot 0 and csc 0 are undefined.
Domain and Period
The domain of the sine and cosine functions is the set of real x values that fall on the function. Remember that r = 1, sin t = y, and cos t = x. As the radius is 1,
-1 = y = 1 -1 = x = 1 Therefore the values of sine and cosine also range between -1 and 1: -1 = sin t = 1 -1 = cos t = 1
Adding 2 to each value of t completes another full revolution around the unit circle. Therefore, the values of sin(t + 2) and cos(t + 2) correspond to those of sin t and cos t. Adding any multiple of 2p, positive or negative, will yield the same results. This means that: sin(t + 2n) = sin t and cos(t + 2n) cos t for any integer n and real number t. Functions that repeat in cycles like this are called periodic. A function is periodic if there is a positive number c such that f(t +c) = f(t) The smallest number c that makes f periodic is called the period of f. It is shifted by c so that after the shift it the same as the original function.