Trigonometric Functions (Coordinate Definition): Let \(\theta\) be an angle in standard position and let \(P(x, y)\) be a point on its terminal side. If \(r = \sqrt{x^2 + y^2}\) is the distance from the origin to \(P\), the six trigonometric functions are defined as the ratios of \(x\), \(y\), and \(r\). For example, \(\sin(\theta) = \frac{y}{r}\) and \(\cos(\theta) = \frac{x}{r}\).
Special Angles: Angles that can be expressed in the form \(\pm 30^\circ k\) or \(\pm 45^\circ k\), where \(k\) is an integer. The exact values of their trigonometric functions can be found using special triangles.
Theorems
Signs of the Trigonometric Functions: The signs of the trigonometric function values depend on the quadrant in which the angle's terminal side lies.
QI: All functions are positive.
QII: Sine and cosecant are positive.
QIII: Tangent and cotangent are positive.
QIV: Cosine and secant are positive.
Trigonometric Ratios are Independent of P: The values of the trigonometric functions of an angle \(\theta\) depend only on the angle itself, not on the specific point \(P(x, y)\) chosen on the terminal side of \(\theta\) (provided the point is not the origin).
Common Mistakes
Confusing Function Notation with Multiplication: The notation \(\cos(\theta)\) represents a single value, which is the output of the cosine function for the input angle \(\theta\). It is not a product of "cos" and "\(\theta\)". The term "cos" by itself has no mathematical meaning.
Incorrectly Stating Undefined Values: Mathematically, it is more accurate to state that a function "is undefined" at a certain value rather than saying it "equals UNDEFINED". For example, the statement "\(\cot(540^\circ)\) is undefined" is preferred over "\(\cot(540^\circ) = \text{UNDEFINED}\)".
Assuming \(r\) Can Be Negative: In the coordinate definition of trigonometric functions, \(r\) represents a distance and is always non-negative.
Ignoring the Quadrant for Signs: When finding trigonometric values, the quadrant of the angle is crucial for determining the correct signs of the \(x\) and \(y\) coordinates. For example, for an obtuse angle (in QII), the \(x\)-coordinate must be negative.