BackTrigonometric Functions of Any Angle: Definitions, Signs, Coterminal Angles, and Reference Angles
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Trigonometric Functions of Any Angle
Definition of the Six Trigonometric Functions
Trigonometric functions can be defined for any angle using the coordinates of a point on the terminal side of the angle in standard position. Let θ be any angle, and let (a, b) be the coordinates of a point (other than the origin) on the terminal side of θ. The distance from the origin to (a, b) is r, where:
The six trigonometric functions are then defined as:
sin θ = \frac{b}{r}
cos θ = \frac{a}{r}
tan θ = \frac{b}{a}
csc θ = \frac{r}{b}
sec θ = \frac{r}{a}
cot θ = \frac{a}{b}
These definitions are valid as long as the denominators are not zero. If a denominator is zero, the corresponding trigonometric function is not defined for that angle.
Trigonometric Functions of Quadrantal Angles
Quadrantal angles are angles whose terminal sides lie along the x- or y-axis (e.g., 0°, 90°, 180°, 270°). The values of the trigonometric functions for these angles are summarized in the table below:
θ (Radians) | θ (Degrees) | sin θ | cos θ | tan θ | csc θ | sec θ | cot θ |
|---|---|---|---|---|---|---|---|
0 | 0° | 0 | 1 | 0 | Not defined | 1 | Not defined |
\frac{\pi}{2} | 90° | 1 | 0 | Not defined | 1 | Not defined | 0 |
\pi | 180° | 0 | -1 | 0 | Not defined | -1 | Not defined |
\frac{3\pi}{2} | 270° | -1 | 0 | Not defined | -1 | Not defined | 0 |
Coterminal Angles
Definition and Properties
Two angles in standard position are called coterminal if they share the same terminal side. Coterminal angles differ by integer multiples of 360° (or 2π radians):
or
where k is any integer.
Examples of Coterminal Angles
390° is coterminal with 30° because 390° = 30° + 360°.
420° is coterminal with 60° because 420° = 60° + 360°.
is coterminal with because .
General Formulas for Coterminal Angles
θ degrees | θ radians |
|---|---|
sin(θ + 360°k) = sin θ | sin(θ + 2πk) = sin θ |
cos(θ + 360°k) = cos θ | cos(θ + 2πk) = cos θ |
tan(θ + 360°k) = tan θ | tan(θ + 2πk) = tan θ |
csc(θ + 360°k) = csc θ | csc(θ + 2πk) = csc θ |
sec(θ + 360°k) = sec θ | sec(θ + 2πk) = sec θ |
cot(θ + 360°k) = cot θ | cot(θ + 2πk) = cot θ |
Signs of Trigonometric Functions in Each Quadrant
Quadrant Sign Rules
The sign of a trigonometric function depends on the quadrant in which the terminal side of the angle lies:
Quadrant of θ | sin θ, csc θ | cos θ, sec θ | tan θ, cot θ |
|---|---|---|---|
I | Positive | Positive | Positive |
II | Positive | Negative | Negative |
III | Negative | Negative | Positive |
IV | Negative | Positive | Negative |
Reference Angles
Definition of Reference Angle
The reference angle for an angle θ is the acute angle formed by the terminal side of θ and the x-axis. Reference angles are always between 0° and 90° (or 0 and π/2 radians).
Examples of Reference Angles
The reference angle for 150° is 30°.
The reference angle for –45° is 45°.
The reference angle for is .
The reference angle for is .
Reference Angle Theorem
If θ is an angle in a quadrant and α is its reference angle, then:
The sign depends on the quadrant in which θ lies.
Examples Using Reference Angles
cos(135°): Reference angle is 45°, and 135° is in quadrant II, so cos(135°) = –cos(45°) = .
tan(600°): Reference angle is 60°, and 600° is in quadrant III, so tan(600°) = tan(60°) = .
Summary: Finding Trigonometric Values for Any Angle
If θ is a quadrantal angle, draw the angle, pick a point on its terminal side, and apply the definitions.
If θ lies in a quadrant:
Find the reference angle α for θ.
Find the value of the trigonometric function at α.
Adjust the sign (+ or –) based on the quadrant in which θ lies.
Additional info:
These notes cover Section 5.4 of a standard Precalculus curriculum, focusing on trigonometric functions of any angle, coterminal angles, quadrant sign rules, and reference angles. The content is essential for understanding trigonometric functions beyond acute angles and is foundational for later topics in trigonometry and calculus.
