Question
i6.4 If \( x=3 \sin \theta \) and \( y=3 \cos \theta \). detennine the value of \( x^{2}+y^{2} \)
Ask by Campbell Sanders. in South Africa
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x^{2} + y^{2} = 9 \)
Solution
1. Substitute the given values into the expression:
\[
x^2 + y^2 = (3\sin \theta)^2 + (3\cos \theta)^2
\]
2. Square the terms:
\[
x^2 + y^2 = 9\sin^2 \theta + 9\cos^2 \theta
\]
3. Factor out the common factor:
\[
x^2 + y^2 = 9(\sin^2 \theta + \cos^2 \theta)
\]
4. Use the Pythagorean identity:
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
5. Substitute the identity into the expression:
\[
x^2 + y^2 = 9 \times 1 = 9
\]
Therefore, the value of \( x^2+y^2 \) is \( 9 \).
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Simplify this solution Beyond the Answer
To find the value of \( x^{2} + y^{2} \), we substitute the given values of \( x \) and \( y \): \[ x^{2} = (3 \sin \theta)^{2} = 9 \sin^{2} \theta \] \[ y^{2} = (3 \cos \theta)^{2} = 9 \cos^{2} \theta \] Now, adding these two results together: \[ x^{2} + y^{2} = 9 \sin^{2} \theta + 9 \cos^{2} \theta = 9 (\sin^{2} \theta + \cos^{2} \theta) \] Since \( \sin^{2} \theta + \cos^{2} \theta = 1 \), we have: \[ x^{2} + y^{2} = 9 \cdot 1 = 9 \] Thus, the value of \( x^{2} + y^{2} \) is \( 9 \).
