Trigonometric Ratios for a Right Triangle - 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

SohCahToa: $$\text{sin} \theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \text{cos} \theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \text{tan} \theta = \frac{\text{opposite}}{\text{adjacent}}$$

Pythagorean Theorem: $$x^2 + y^2 = z^2$$

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

3. Examples

Example 1 (Basic)

Problem: Given a right triangle with an angle $$\theta$$ where the opposite side is 3 units and the adjacent side is 4 units, find $$\text{tan} \theta$$.

Step-by-Step Solution:

Identify the opposite and adjacent sides: Opposite = 3, Adjacent = 4.
Calculate $$\text{tan} \theta$$: $$\text{tan} \theta = \frac{3}{4}$$.

Validation: Substitute values → Original: tan(θ) = 3/4; Simplified: 0.75 ✓

Example 2 (Intermediate)

Problem: A ladder leaning against a wall makes an angle of $$60^\circ$$ with the ground. If the ladder is 10 meters long, how high up the wall does it reach?

Step-by-Step Solution:

Identify the given information: Angle = $$60^\circ$$, Hypotenuse = 10 meters.
Use sine function: $$\text{sin} 60^\circ = \frac{\text{opposite}}{10}$$.
Calculate the height: $$\text{sin} 60^\circ = \frac{\sqrt{3}}{2}$$, so $$\frac{\sqrt{3}}{2} = \frac{\text{opposite}}{10}$$.
Solve for opposite: $$\text{opposite} = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3}$$ meters.

Validation: Substitute values → Original: sin(60°) = √3/2; Simplified: 5√3 ≈ 8.66 meters ✓

4. Problem-Solving Techniques

Visual Strategy: Draw a right triangle and label all known sides and angles.
Error-Proofing: Double-check trigonometric identities and Pythagorean theorem applications.
Concept Reinforcement: Practice with real-world scenarios to understand practical applications.