Trigonometric Functions: Definitions, Properties, and Quadrantal Angles

Trigonometric Functions and Their Foundations

Pythagorean Theorem

The Pythagorean Theorem is a fundamental result in geometry, especially relevant in trigonometry for analyzing right triangles. It states that in any right triangle, the sum of the squares of the lengths of the two legs equals the square of the length of the hypotenuse.

• Formula:

$a^2 + b^2 = c^2$

• Where: a and b are the legs, and c is the hypotenuse.

• Application: Used to relate the sides of right triangles, which is foundational for defining trigonometric functions.

Distance Formula

The Distance Formula calculates the distance between two points in the coordinate plane. It is derived from the Pythagorean Theorem and is essential for defining the radius in trigonometric contexts.

• Formula:

$d(P, Q) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Application: Used to find the length r from the origin to a point P(x, y) on the terminal side of an angle.

Definition of the Six Trigonometric Functions

Trigonometric Functions in the Coordinate Plane

Trigonometric functions generalize the relationships between angles and side lengths in right triangles to any angle in the coordinate plane. For an angle θ in standard position, and a point P(x, y) (not the origin) on its terminal side, the distance from the origin is:

$r = \sqrt{x^2 + y^2}$

• Note: r is always positive and represents the radius or distance from the origin to the point.

Definitions of the Six Trigonometric Functions

The six trigonometric functions are defined as ratios involving x, y, and r:

• Sine (sin): $\sin \theta = \dfrac{y}{r}$

• Cosecant (csc): $\csc \theta = \dfrac{r}{y}$,   y ≠ 0

• Cosine (cos): $\cos \theta = \dfrac{x}{r}$

• Secant (sec): $\sec \theta = \dfrac{r}{x}$,   x ≠ 0

• Tangent (tan): $\tan \theta = \dfrac{y}{x}$,   x ≠ 0

• Cotangent (cot): $\cot \theta = \dfrac{x}{y}$,   y ≠ 0

Example: If P(3, 4) is on the terminal side of θ, then r = 5 (by the Pythagorean Theorem), so $\sin \theta = \dfrac{4}{5}$, $\cos \theta = \dfrac{3}{5}$, $\tan \theta = \dfrac{4}{3}$, etc.

Quadrantal Angles

Definition and Properties

Quadrantal angles are angles whose terminal sides lie along the x-axis or y-axis (i.e., at 0°, 90°, 180°, or 270°, or equivalently 0, $\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2}$ radians). For these angles, the coordinates of points one unit from the origin (on the unit circle) are:

• (1, 0)   (x = 1, y = 0, r = 1)

• (0, 1)   (x = 0, y = 1, r = 1)

• (-1, 0)   (x = -1, y = 0, r = 1)

• (0, -1)   (x = 0, y = -1, r = 1)

For quadrantal angles, some trigonometric functions are undefined because their denominators become zero.

Undefined Function Values for Quadrantal Angles

• If x = 0 (terminal side on y-axis): tan θ and sec θ are undefined.

• If y = 0 (terminal side on x-axis): cot θ and csc θ are undefined.

Example: At θ = 90° (point (0, 1)), $\tan 90^\circ = \dfrac{1}{0}$ (undefined), $\sec 90^\circ = \dfrac{1}{0}$ (undefined), but $\sin 90^\circ = 1$, $\cos 90^\circ = 0$.

Table: Trigonometric Function Values at Quadrantal Angles

Angle (θ)	Coordinates (x, y)	sin θ	cos θ	tan θ	csc θ	sec θ	cot θ
0° (0)	(1, 0)	0	1	0	undefined	1	undefined
90° ($\frac{\pi}{2}$)	(0, 1)	1	0	undefined	1	undefined	0
180° ($\pi$)	(-1, 0)	0	-1	0	undefined	-1	undefined
270° ($\frac{3\pi}{2}$)	(0, -1)	-1	0	undefined	-1	undefined	0

Additional info: Table values inferred from standard trigonometric definitions and the unit circle.

Calculator Use in Trigonometry

Degree and Radian Modes

When using calculators for trigonometric calculations, ensure the calculator is set to the correct mode:

• Degree mode: For angles measured in degrees (e.g., 30°, 90°).

• Radian mode: For angles measured in radians (e.g., $\frac{\pi}{6}$, $\frac{\pi}{2}$).

Calculators can be used to explore and verify properties of trigonometric functions, but formal proofs are required for rigorous understanding.

Summary

• The six trigonometric functions are defined for any angle using coordinates and the distance from the origin.

• Quadrantal angles have special properties, with some function values undefined due to division by zero.

• Understanding these definitions is foundational for further study in trigonometry, including the unit circle and graphing trigonometric functions.