If x = p sin theater and y = p cos theater show that x square + y square = p square .
To prove that x^2 + y^2 = p^2, we simply need to substitute the given values of x and y in terms of p and theta into the equation and simplify:
x^2 + y^2 = (p sin(theta))^2 + (p cos(theta))^2
= p^2(sin^2(theta) + cos^2(theta))
= p^2(1) (since sin^2(theta) + cos^2(theta) = 1)
= p^2
Therefore, we have shown that x^2 + y^2 = p^2, which is the desired result.
To prove that x^2 + y^2 = p^2, we can start by substituting the given equations for x and y into the expression:
x^2 + y^2 = (p sin(theta))^2 + (p cos(theta))^2
Using the trigonometric identity sin^2(theta) + cos^2(theta) = 1, we can simplify the equation as follows:
x^2 + y^2 = p^2 sin^2(theta) + p^2 cos^2(theta)
Next, we can factor out the common term "p^2" from both terms:
x^2 + y^2 = p^2 (sin^2(theta) + cos^2(theta))
Applying the trigonometric identity sin^2(theta) + cos^2(theta) = 1 once again, we can simplify the equation further:
x^2 + y^2 = p^2 (1)
Finally, simplifying the right-hand side of the equation, we get:
x^2 + y^2 = p^2
Therefore, x^2 + y^2 is equal to p^2, as we set out to prove.