Explain why sin(x) = 1 - cos(x) is not an identity.
Understand the Problem
The question is asking why the equation sin(x) = 1 - cos(x) is not a trigonometric identity. This means we need to analyze both sides of the equation and determine if they are equivalent for all values of x. The high-level approach will involve examining the definitions and properties of the sine and cosine functions.
Answer
sin(x) = 1 - cos(x) is not an identity because it's not true for all x; try x = π/4: sin(π/4) ≠ 1 - cos(π/4).
To show that sin(x) = 1 - cos(x) is not an identity, we can find a specific value of x where the two sides of the equation do not equal each other. For example, if x = π/4, then sin(π/4) = √2/2, but 1 - cos(π/4) = 1 - √2/2, which are not equal. Therefore, the equation is not an identity because it does not hold for this value of x.
Answer for screen readers
To show that sin(x) = 1 - cos(x) is not an identity, we can find a specific value of x where the two sides of the equation do not equal each other. For example, if x = π/4, then sin(π/4) = √2/2, but 1 - cos(π/4) = 1 - √2/2, which are not equal. Therefore, the equation is not an identity because it does not hold for this value of x.
More Information
An identity in trigonometry is an equation that holds true for all values of x, but sin(x) = 1 - cos(x) only holds for specific values, thus failing to be an identity.
Tips
A common mistake is neglecting to check multiple values of x to confirm if an equation holds universally. Always test with different values to ensure consistency.
Sources
- Prove that the given equation is not an identity by finding x for which ... - homework.study.com
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