Find Trig Ratio From a Point - Algebra-2

1. Fundamental Concepts

Definition: Trigonometric ratios are relationships between the sides of a right triangle and its angles.
Sine (sin): The ratio of the length of the opposite side to the hypotenuse.
Cosine (cos): The ratio of the length of the adjacent side to the hypotenuse.
Tangent (tan): The ratio of the length of the opposite side to the adjacent side.

2. Key Concepts

Basic Rule: $${\text{For a point } (x, y) \text{ on the unit circle:}}$$ $${\sin(\theta) = \frac{y}{r}, \quad \cos(\theta) = \frac{x}{r}, \quad \tan(\theta) = \frac{y}{x}}$$

Degree Preservation: The trigonometric ratios depend on the angle $\theta$ and not on the size of the triangle.

Application: Used in various fields including physics, engineering, and navigation.

3. Examples

Example 1 (Basic)

Problem: Find the sine, cosine, and tangent of the angle $\theta$ for the point $(3, 4)$ on the coordinate plane.

Step-by-Step Solution:

Calculate the radius $r$: $r = \sqrt{x^2 + y^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.
Find $\sin(\theta)$, $\cos(\theta)$, and $\tan(\theta)$:
- $\sin(\theta) = \frac{y}{r} = \frac{4}{5}$
- $\cos(\theta) = \frac{x}{r} = \frac{3}{5}$
- $\tan(\theta) = \frac{y}{x} = \frac{4}{3}$

Validation: Substitute $x = 3$, $y = 4$, $r = 5$. Original values match calculated values.

Example 2 (Intermediate)

Problem: Given the point $(-2, -2)$, find the trigonometric ratios for the corresponding angle $\theta$.

Step-by-Step Solution:

Calculate the radius $r$: $r = \sqrt{x^2 + y^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$.
Find $\sin(\theta)$, $\cos(\theta)$, and $\tan(\theta)$:
- $\sin(\theta) = \frac{y}{r} = \frac{-2}{2\sqrt{2}} = -\frac{1}{\sqrt{2}} = -\frac{\sqrt{2}}{2}$
- $\cos(\theta) = \frac{x}{r} = \frac{-2}{2\sqrt{2}} = -\frac{1}{\sqrt{2}} = -\frac{\sqrt{2}}{2}$
- $\tan(\theta) = \frac{y}{x} = \frac{-2}{-2} = 1$

Validation: Substitute $x = -2$, $y = -2$, $r = 2\sqrt{2}$. Original values match calculated values.

4. Problem-Solving Techniques

Visual Strategy: Use the unit circle to visualize the trigonometric ratios.
Error-Proofing: Always check the quadrant to determine the sign of the trigonometric ratios.
Concept Reinforcement: Practice with different points to understand the relationship between the coordinates and the trigonometric ratios.