What is 1 - cos(x) equal to?
Understand the Problem
The question is asking for the mathematical expression of 1 minus the cosine of x (1 - cos(x)). This typically relates to trigonometric identities and is often simplified or used in calculus and physics.
Answer
$2\sin^2\left(\frac{x}{2}\right)$
Answer for screen readers
The mathematical expression of $1 - \cos(x)$ is $2\sin^2\left(\frac{x}{2}\right)$.
Steps to Solve
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Identify the expression
The expression we need to work with is $1 - \cos(x)$. -
Understanding trigonometric identities
This expression can often be simplified using the half-angle identity in trigonometry.
The identity states:
$$ 1 - \cos(x) = 2\sin^2\left(\frac{x}{2}\right) $$ -
Apply the identity
Replace $1 - \cos(x)$ with the identity we identified:
$$ 1 - \cos(x) = 2\sin^2\left(\frac{x}{2}\right) $$ -
Conclusion
The final expression can be summarized with the trigonometric identity applied.
The mathematical expression of $1 - \cos(x)$ is $2\sin^2\left(\frac{x}{2}\right)$.
More Information
This expression is useful in various applications, including solving triangles and integrals in calculus. It's part of the Pythagorean identities and can also help in simplifying integrals involving trigonometric functions.
Tips
- Forgetting to apply the half-angle identity properly. Always check if the expression can be simplified using known trigonometric identities.
- Misinterpreting the expression; sometimes it's easy to overlook the significance of the angle divisions.
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