BackTrigonometric Functions and Fundamental Identities (Precalculus Study Notes)
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Trigonometric Functions and Fundamental Identities
Introduction to Trigonometric Functions
Trigonometric functions are fundamental in the study of angles and their relationships to points on the coordinate plane. These functions are essential in Precalculus and have wide applications in science and engineering.
Six Trigonometric Functions: Sine (sin), Cosine (cos), Tangent (tan), Cotangent (cot), Secant (sec), and Cosecant (csc).
Standard Position: An angle is in standard position if its vertex is at the origin and its initial side lies along the positive x-axis.
Reference Point: For any angle θ, choose a point P(x, y) on its terminal side (other than the origin). The distance from the origin to P is , where .
Definitions of the Six Trigonometric Functions
The trigonometric functions are defined as follows for a point (x, y) on the terminal side of angle θ:
(x ≠ 0)
(y ≠ 0)
(x ≠ 0)
(y ≠ 0)
Examples: Finding Trigonometric Function Values
Example 1: If the terminal side of θ passes through (–3, –4): , , , ,
Example 2: For the line , choose , , :
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°, 360°).
Example: For 90°, using (0, 1): , , is undefined , is undefined,
Example: For (–3, 0): , , is undefined, , is undefined
Conditions for Undefined Function Values
If the terminal side lies on the y-axis: tangent and secant are undefined.
If the terminal side lies on the x-axis: cotangent and cosecant are undefined.
Table: Function Values of Quadrantal Angles
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 |
360° | 0 | 1 | 0 | Undefined | 1 | Undefined |
Reciprocal Identities
Reciprocal identities relate each trigonometric function to its reciprocal:
Example: Using Reciprocal Identities
Given ,
Given ,
Signs and Ranges of Trigonometric Functions
The sign of each trigonometric function depends on the quadrant in which the terminal side of the angle lies.
θ in Quadrant | sin θ | cos θ | tan θ | cot θ | sec θ | csc θ |
|---|---|---|---|---|---|---|
I | + | + | + | + | + | + |
II | + | – | – | – | – | + |
III | – | – | + | + | – | – |
IV | – | + | – | – | + | – |
Ranges of Trigonometric Functions
Trigonometric Function of θ | Range (Set-Builder Notation) | Range (Interval Notation) |
|---|---|---|
sin θ, cos θ | |y| ≤ 1 | [–1, 1] |
tan θ, cot θ | y is a real number | (–∞, ∞) |
sec θ, csc θ | |y| ≥ 1 | (–∞, –1] ∪ [1, ∞) |
Example: Deciding Whether a Value Is in the Range
is impossible, since
is possible, since tangent can be any real number
is impossible, since
Pythagorean Identities
Pythagorean identities are derived from the Pythagorean Theorem and are fundamental in trigonometry:
Quotient Identities
Quotient identities express tangent and cotangent in terms of sine and cosine:
Examples: Using Identities and Quadrant Information
Given: , θ in quadrant III. (negative in quadrant III) (negative in quadrant III)
Expressing sec θ in terms of sin θ (θ in quadrant IV): (positive in quadrant IV)
Applications: Grade Resistance Example
Problem: A downhill highway grade is modeled by in quadrant IV. Find the grade and the grade resistance for a 3000-pound car.
Solution: Slope is 0.06, so grade is –6%. For a 100 ft run, , Grade resistance pounds (negative indicates downhill force).
