Trigonometric Functions of Any Angle: Definitions, Signs, Coterminal Angles, and Reference Angles

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:

$r = \sqrt{a^2 + b^2}$

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):

$\theta + 360^\circ k$ or $\theta + 2\pi k$

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°.

• $\frac{9\pi}{4}$ is coterminal with $\frac{\pi}{4}$ because $\frac{9\pi}{4} = \frac{\pi}{4} + 2\pi$.

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 $\frac{9\pi}{4}$ is $\frac{\pi}{4}$.

• The reference angle for $-\frac{5\pi}{6}$ is $\frac{\pi}{6}$.

Reference Angle Theorem

If θ is an angle in a quadrant and α is its reference angle, then:

• $\sin \theta = \pm \sin \alpha$

• $\cos \theta = \pm \cos \alpha$

• $\tan \theta = \pm \tan \alpha$

• $\csc \theta = \pm \csc \alpha$

• $\sec \theta = \pm \sec \alpha$

• $\cot \theta = \pm \cot \alpha$

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°) = $-\frac{\sqrt{2}}{2}$.

• tan(600°): Reference angle is 60°, and 600° is in quadrant III, so tan(600°) = tan(60°) = $\sqrt{3}$.

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:

  1. Find the reference angle α for θ.

  2. Find the value of the trigonometric function at α.

  3. 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.