๐ Understanding Trigonometric Functions with a Point (x, y)
Trigonometric functions relate angles of a right triangle to the ratios of its sides. When you're given a point (x, y), you can think of it as a point on the coordinate plane that forms a right triangle with the x-axis. We'll use this triangle to define all six trig functions.
๐ A Brief History
The study of trigonometry dates back to ancient civilizations like the Egyptians, Babylonians, and Greeks. Early astronomers used trigonometric relationships to calculate the positions of stars and planets. Over time, these concepts were formalized and expanded upon, leading to the trigonometric functions we use today. Hipparchus of Nicaea is credited with the initial development of trigonometry, followed by significant contributions from mathematicians like Ptolemy and Indian scholars who refined sine and cosine functions.
๐ Key Principles
- ๐ Right Triangle Formation: Given a point (x, y), visualize a right triangle formed by the x-axis, a vertical line from (x, y) to the x-axis, and the line segment connecting the origin (0, 0) to (x, y).
- ๐ Defining 'r': Calculate the length of the hypotenuse (r) using the Pythagorean theorem: $r = \sqrt{x^2 + y^2}$. This 'r' represents the distance from the origin to the point (x, y).
- โ๏ธ Sine (sin ฮธ): Defined as the ratio of the opposite side (y) to the hypotenuse (r): $\sin(\theta) = \frac{y}{r}$.
- โ๏ธ Cosine (cos ฮธ): Defined as the ratio of the adjacent side (x) to the hypotenuse (r): $\cos(\theta) = \frac{x}{r}$.
- โ๏ธ Tangent (tan ฮธ): Defined as the ratio of the opposite side (y) to the adjacent side (x): $\tan(\theta) = \frac{y}{x}$.
- โ๏ธ Cosecant (csc ฮธ): The reciprocal of sine: $\csc(\theta) = \frac{r}{y}$.
- โ๏ธ Secant (sec ฮธ): The reciprocal of cosine: $\sec(\theta) = \frac{r}{x}$.
- โ๏ธ Cotangent (cot ฮธ): The reciprocal of tangent: $\cot(\theta) = \frac{x}{y}$.
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Step-by-Step Guide
- ๐ Step 1: Identify (x, y): Note the coordinates of the given point.
- ๐ Step 2: Calculate 'r': Use the formula $r = \sqrt{x^2 + y^2}$ to find the distance from the origin.
- โ๏ธ Step 3: Apply the Definitions: Use the definitions of sine, cosine, tangent, cosecant, secant, and cotangent to calculate their values.
๐งช Real-world Examples
Example 1:
Given the point (3, 4):
- ๐ $x = 3$, $y = 4$
- ๐ $r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$
| Function |
Calculation |
Value |
| $\sin(\theta)$ |
$\frac{y}{r} = \frac{4}{5}$ |
0.8 |
| $\cos(\theta)$ |
$\frac{x}{r} = \frac{3}{5}$ |
0.6 |
| $\tan(\theta)$ |
$\frac{y}{x} = \frac{4}{3}$ |
1.33 |
| $\csc(\theta)$ |
$\frac{r}{y} = \frac{5}{4}$ |
1.25 |
| $\sec(\theta)$ |
$\frac{r}{x} = \frac{5}{3}$ |
1.67 |
| $\cot(\theta)$ |
$\frac{x}{y} = \frac{3}{4}$ |
0.75 |
Example 2:
Given the point (-5, 12):
- ๐ $x = -5$, $y = 12$
- ๐ $r = \sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$
| Function |
Calculation |
Value |
| $\sin(\theta)$ |
$\frac{y}{r} = \frac{12}{13}$ |
0.92 |
| $\cos(\theta)$ |
$\frac{x}{r} = \frac{-5}{13}$ |
-0.38 |
| $\tan(\theta)$ |
$\frac{y}{x} = \frac{12}{-5}$ |
-2.4 |
| $\csc(\theta)$ |
$\frac{r}{y} = \frac{13}{12}$ |
1.08 |
| $\sec(\theta)$ |
$\frac{r}{x} = \frac{13}{-5}$ |
-2.6 |
| $\cot(\theta)$ |
$\frac{x}{y} = \frac{-5}{12}$ |
-0.42 |
๐ Conclusion
By understanding the relationship between a point (x, y) and the right triangle it forms, you can easily define all six trigonometric functions. Remember to calculate 'r' using the Pythagorean theorem and then apply the definitions of sine, cosine, tangent, cosecant, secant, and cotangent. With practice, you'll master these concepts in no time!