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Let x= cos theta and y = sin theta for a...

Let `x= cos theta and y = sin theta` for any real value `theta`. Then `x^(2)+y^(2)=`

A

`-1`

B

0

C

1

D

It cannot be determined from the information given

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to show that \( x^2 + y^2 = 1 \) given that \( x = \cos \theta \) and \( y = \sin \theta \). ### Step-by-Step Solution: 1. **Substitute the values of \( x \) and \( y \)**: \[ x = \cos \theta \quad \text{and} \quad y = \sin \theta \] 2. **Write the expression \( x^2 + y^2 \)**: \[ x^2 + y^2 = (\cos \theta)^2 + (\sin \theta)^2 \] 3. **Use the Pythagorean identity**: According to the Pythagorean identity in trigonometry, we have: \[ \cos^2 \theta + \sin^2 \theta = 1 \] 4. **Substitute the identity into the expression**: \[ x^2 + y^2 = \cos^2 \theta + \sin^2 \theta = 1 \] 5. **Conclusion**: Therefore, we conclude that: \[ x^2 + y^2 = 1 \] ### Final Answer: \[ x^2 + y^2 = 1 \]
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