Trigonometric Functions: Definitions, Applications, and Quadrantal Angles

Trigonometric Functions

Introduction to Trigonometric Functions

Trigonometric functions are fundamental mathematical tools used to relate the angles of a triangle to the lengths of its sides. In the context of the coordinate plane, these functions are defined using a point P(x, y) on the terminal side of an angle θ in standard position, where the vertex is at the origin and the initial side lies along the positive x-axis.

• Standard Position: An angle is in standard position if its vertex is at the origin and its initial side is along the positive x-axis.

• Terminal Side: The ray that defines the angle's measure from the initial side.

The Pythagorean Theorem and the Distance Formula

The distance from the origin to the point P(x, y) is given by the Pythagorean Theorem:

• Distance Formula: $r = \sqrt{x^2 + y^2}$ where r is the distance from the origin to the point (x, y).

Definitions of the Six Trigonometric Functions

The six trigonometric functions of an angle θ are defined as follows, using the coordinates of P(x, y) and the distance r:

• Sine: $\sin \theta = \frac{y}{r}$

• Cosine: $\cos \theta = \frac{x}{r}$

• Tangent: $\tan \theta = \frac{y}{x}, \quad x \neq 0$

• Cosecant: $\csc \theta = \frac{r}{y}, \quad y \neq 0$

• Secant: $\sec \theta = \frac{r}{x}, \quad x \neq 0$

• Cotangent: $\cot \theta = \frac{x}{y}, \quad y \neq 0$

Finding Trigonometric Function Values

Example 1: Point in the First Quadrant

Given the terminal side of angle θ passes through the point (8, 15), find the six trigonometric functions.

• Step 1: Calculate r: $r = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17$

• Step 2: Substitute into the definitions: $\sin \theta = \frac{15}{17}$ $\cos \theta = \frac{8}{17}$ $\tan \theta = \frac{15}{8}$ $\csc \theta = \frac{17}{15}$ $\sec \theta = \frac{17}{8}$ $\cot \theta = \frac{8}{15}$

Example 2: Point in the Third Quadrant

Given the terminal side of angle θ passes through the point (-3, -4), find the six trigonometric functions.

• Step 1: Calculate r: $r = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

• Step 2: Substitute into the definitions: $\sin \theta = \frac{-4}{5}$ $\cos \theta = \frac{-3}{5}$ $\tan \theta = \frac{-4}{-3} = \frac{4}{3}$ $\csc \theta = \frac{5}{-4} = -\frac{5}{4}$ $\sec \theta = \frac{5}{-3} = -\frac{5}{3}$ $\cot \theta = \frac{-3}{-4} = \frac{3}{4}$

Quadrantal Angles

Definition and Properties

Quadrantal angles are angles whose terminal sides lie along the x-axis or y-axis (e.g., 0°, 90°, 180°, 270°). The values of trigonometric functions for these angles often result in 0, 1, or are undefined due to division by zero.

• Terminal Side on y-axis: Tangent and secant functions are undefined.

• Terminal Side on x-axis: Cotangent and cosecant functions are undefined.

Function Values for Quadrantal Angles

Angle (θ)	sin θ	cos θ	tan θ	cot θ	sec θ	csc θ
0°	0	1	0	undefined	1	undefined
90°	1	0	undefined	0	undefined	1
180°	0	-1	0	undefined	-1	undefined
270°	-1	0	undefined	0	undefined	-1

Using a Calculator for Trigonometric Functions

Calculator Settings and Common Errors

When evaluating trigonometric functions using a calculator, it is essential to ensure the calculator is set to the correct angle measure (degree or radian mode). Most trigonometry problems in introductory courses use degrees.

• Degree Mode: Set your calculator to degree mode for angles measured in degrees.

• Common Error: Using radian mode when degrees are required can lead to incorrect results.

Example: Calculating sin(90°)

• Set calculator to degree mode.

• Input: sin(90)

• Result: 1

Summary Table: Trigonometric Function Definitions

Function	Definition	Domain Restriction
sin θ	$\frac{y}{r}$	r ≠ 0
cos θ	$\frac{x}{r}$	r ≠ 0
tan θ	$\frac{y}{x}$	x ≠ 0
csc θ	$\frac{r}{y}$	y ≠ 0
sec θ	$\frac{r}{x}$	x ≠ 0
cot θ	$\frac{x}{y}$	y ≠ 0

Additional info: The examples and tables above are expanded for clarity and completeness, including domain restrictions and typical calculator usage advice for trigonometric functions.