Trig Ratios - Physics

1. Fundamental Concepts

Definition: Trigonometric ratios are the 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: $${\text{{S}}}\text{{in}} \theta = \frac{\text{{opposite}}}{\text{{hypotenuse}}}$$ $${\text{{C}}}\text{{os}} \theta = \frac{\text{{adjacent}}}{\text{{hypotenuse}}}$$ $${\text{{T}}}\text{{an}} \theta = \frac{\text{{opposite}}}{\text{{adjacent}}}$$

Pythagorean Identity: $${\text{{sin}}}^2 \theta + {\text{{cos}}}^2 \theta = 1$$

Application: Used in physics for resolving vectors into components

3. Examples

Example 1 (Basic)

Problem: Find $$\text{{sin}} \theta$$ and $$\text{{cos}} \theta$$ if the opposite side is 3 and the hypotenuse is 5.

Step-by-Step Solution:

Calculate $$\text{{sin}} \theta$$: $$\text{{sin}} \theta = \frac{3}{5}$$
Calculate $$\text{{cos}} \theta$$: $$\text{{cos}} \theta = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \frac{4}{5}$$

Validation: Substitute values → $$\text{{sin}}^2 \theta + \text{{cos}}^2 \theta = \left(\frac{3}{5}\right)^2 + \left(\frac{4}{5}\right)^2 = 1$$ ✓

Example 2 (Intermediate)

Problem: A ladder 10 meters long leans against a wall, forming an angle of 60 degrees with the ground. Find the height it reaches on the wall.

Step-by-Step Solution:

Identify the relevant trigonometric ratio: $$\text{{sin}} 60^\circ = \frac{\text{{height}}}{10}$$
Use the value of $$\text{{sin}} 60^\circ = \frac{\sqrt{3}}{2}$$: $$\frac{\sqrt{3}}{2} = \frac{\text{{height}}}{10}$$
Solve for height: $$\text{{height}} = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3} \text{{ meters}}$$

Validation: Substitute values → $$\text{{sin}} 60^\circ = \frac{5\sqrt{3}}{10} = \frac{\sqrt{3}}{2}$$ ✓

4. Problem-Solving Techniques

Visual Strategy: Draw diagrams to visualize the problem and label all known quantities.
Error-Proofing: Double-check units and ensure that all trigonometric functions are used correctly.
Concept Reinforcement: Practice with a variety of problems to reinforce understanding of trigonometric ratios.