How do you evaluate the expression 1 – cos(θ) ?

by

The expression 1 – cos(θ) can be evaluated using various methods depending on the context in which it is used. Here’s a breakdown of how you can approach this expression:

1. Understanding the Expression

The term cos(θ) represents the cosine of an angle θ, and the expression 1 – cos(θ) is simply the difference between 1 and the cosine of that angle.

2. Values of Cosine

The value of cos(θ) varies depending on the measure of θ. For example:

  • If θ = 0°, then cos(0°) = 1, thus 1 – cos(0°) = 0.
  • If θ = 60°, then cos(60°) = 0.5, so 1 – cos(60°) = 0.5.
  • If θ = 90°, then cos(90°) = 0, resulting in 1 – cos(90°) = 1.
  • If θ = 180°, then cos(180°) = -1, leading to 1 – cos(180°) = 2.

3. Applications

This expression is often used in various fields such as physics, engineering, and mathematics, particularly in problems involving waves, oscillations, and circular motion. It can also be related to the Pythagorean identity:

sin²(θ) + cos²(θ) = 1, where we can express 1 – cos(θ) in terms of sine:

1 – cos(θ) = sin²(θ/2).

4. Conclusion

In summary, to evaluate the expression 1 – cos(θ), substitute the value of θ to find its specific value. Additionally, understanding its applications and connection to other trigonometric identities enriches the evaluation process.

Leave a Comment