Question
Use an identity to reduce the power of the trigonometric function to a trigonometric function raised to the first power. 71. \( \sin ^{2} x= \)
Ask by George Reese. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( \sin^2 x = 1 - \cos^2 x \)
Solution
To reduce the power of the trigonometric function \( \sin^2 x \) to a first power, we can use the Pythagorean identity. The relevant identity is:
\[
\sin^2 x = 1 - \cos^2 x
\]
This identity allows us to express \( \sin^2 x \) in terms of \( \cos^2 x \).
Thus, we can write:
\[
\sin^2 x = 1 - \cos^2 x
\]
This is the reduced form of \( \sin^2 x \) using a trigonometric identity.
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Simplify this solution Bonus Knowledge
To reduce \( \sin^2 x \), we can use the Pythagorean identity, which states \( \sin^2 x + \cos^2 x = 1 \). By rearranging this identity, we find that \( \sin^2 x = 1 - \cos^2 x \). This transformation simplifies the power of the sine function into a more manageable form involving cosine. Additionally, it's helpful to remember that when working with power-reduction formulas, other identities like the double angle formulas can come in handy. For instance, if you're looking to express \( \sin^2 x \) in terms of the angle itself, you could also use the identity \( \sin^2 x = \frac{1 - \cos(2x)}{2} \). This gives you a first-power trigonometric function of \( x \) in a neat form!
