Cos 2t is a trigonometric expression that can be expanded using various trigonometric identities. Here are some common ways to express cos 2t:
Double Angle Formula
The most common way to express cos 2t is using the double angle formula:
cos 2t = cos²t - sin²t
This formula directly relates the cosine of twice an angle to the squares of the cosine and sine of the original angle.
Alternative Forms
The double angle formula can be further manipulated to obtain alternative forms:
- cos 2t = 2cos²t - 1
- cos 2t = 1 - 2sin²t
These alternative forms can be useful depending on the context of the problem.
Practical Applications
The expression cos 2t appears in various applications, including:
- Physics: Describing oscillations and wave phenomena.
- Engineering: Analyzing periodic signals and systems.
- Mathematics: Solving trigonometric equations and simplifying expressions.
Example
Let's say we want to find the value of cos 60° using the double angle formula:
- We know that cos 30° = √3/2.
- Using the formula cos 2t = 2cos²t - 1, we can find cos 60°:
- cos 60° = 2(√3/2)² - 1
- cos 60° = 2(3/4) - 1
- cos 60° = 1/2
Therefore, cos 60° is equal to 1/2.
