4-4 Trigonometry Explained
Key Concepts of Trigonometry
Trigonometry is the study of relationships involving lengths and angles of triangles. Key concepts include:
- Sine (sin): The ratio of the length of the opposite side to the length of the hypotenuse in a right triangle.
- Cosine (cos): The ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.
- Tangent (tan): The ratio of the length of the opposite side to the length of the adjacent side in a right triangle.
- Trigonometric Identities: Equations that are true for all values of the variables involved.
1. Sine (sin)
The sine of an angle in a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse. The sine function is defined for angles between 0 and 180 degrees.
Example:
For a right triangle with an angle θ, opposite side length 3 units, and hypotenuse length 5 units, the sine of θ is \( \sin(\theta) = \frac{3}{5} \).
2. Cosine (cos)
The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse. The cosine function is also defined for angles between 0 and 180 degrees.
Example:
For a right triangle with an angle θ, adjacent side length 4 units, and hypotenuse length 5 units, the cosine of θ is \( \cos(\theta) = \frac{4}{5} \).
3. Tangent (tan)
The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. The tangent function is defined for angles between 0 and 180 degrees, except for 90 degrees where it is undefined.
Example:
For a right triangle with an angle θ, opposite side length 3 units, and adjacent side length 4 units, the tangent of θ is \( \tan(\theta) = \frac{3}{4} \).
4. Trigonometric Identities
Trigonometric identities are equations that are true for all values of the variables involved. Some common identities include:
- Pythagorean Identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
- Tangent Identity: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)
- Cofunction Identities: \( \sin(90^\circ - \theta) = \cos(\theta) \) and \( \cos(90^\circ - \theta) = \sin(\theta) \)
Example:
Using the Pythagorean identity, if \( \sin(\theta) = \frac{3}{5} \), then \( \cos(\theta) = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \sqrt{\frac{16}{25}} = \frac{4}{5} \).
Examples and Analogies
To better understand trigonometry, consider the following analogy:
Imagine a right triangle as a ladder leaning against a wall. The sine of the angle the ladder makes with the ground tells you how steep the ladder is. The cosine tells you how far the ladder is from the wall. The tangent tells you the slope of the ladder.
Practical Applications
Understanding trigonometry is crucial for various real-world applications, such as:
- Engineering for calculating forces and stresses in structures.
- Physics for analyzing motion and waves.
- Navigation for determining positions and directions.