The content you provided consists of fundamental trigonometric identities that connect Cartesian coordinates (x, y) to polar coordinates (r, θ). Below is a clear explanation of each identity and their relationships:

Key Background

For any point ((x, y)) in the plane (not at the origin):

• (r) = radial distance from the origin to ((x, y)) (hypotenuse of a right triangle with legs (x) and (y)).
• (\theta) = angle between the positive (x)-axis and the line from the origin to ((x, y)).

Identity Explanations

1. Pythagorean Theorem: (x^2 + y^2 = r^2)
   From the right triangle formed by ((0,0)), ((x,0)), and ((x,y)), the hypotenuse (r = \sqrt{x^2 + y^2}). Squaring both sides gives this identity.

2. Cartesian to Polar (x): (x = r \cos\theta)
   In the right triangle, (\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{r}). Rearranged: (x = r \cos\theta).

3. Cartesian to Polar (y): (y = r \sin\theta)
   Similarly, (\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{r}). Rearranged: (y = r \sin\theta).

4. Tangent of θ: (\tan\theta = \frac{y}{x})
   (\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{y/r}{x/r} = \frac{y}{x}) (valid if (x \neq 0)).

5. Sine of θ: (\sin\theta = \frac{y}{r})
   Directly from identity 3 (divide both sides by (r)).

6. Cosine of θ: (\cos\theta = \frac{x}{r})
   Directly from identity 2 (divide both sides by (r)).

7. y in terms of x and θ: (y = x \tan\theta)
   Substitute (\tan\theta = \frac{y}{x}) into the equation: (y = x \cdot \frac{y}{x} = y) (valid if (x \neq 0)).

8. x in terms of y and θ: (x = y \cot\theta)
   (\cot\theta = \frac{1}{\tan\theta} = \frac{x}{y}). Rearranged: (x = y \cdot \frac{x}{y} = x) (valid if (y \neq 0)).

These identities are essential for converting between Cartesian and polar coordinate systems, solving trigonometric problems, and analyzing geometric shapes (like circles, spirals, etc.) in polar form. If you have a specific problem or question related to these identities, feel free to share it!

Final Takeaway: All these equations are consistent and derived from basic trigonometry and the Pythagorean theorem. They form the core of coordinate system conversions.
(\boxed{\text{These are fundamental Cartesian-polar coordinate identities}}) (if a boxed answer is needed, but since it's explanatory, this summary works).
But if you need a specific identity verified, let me know! For example, verifying (y = x \tan\theta):
From (x = r\cosθ) and (y = r\sinθ), (\frac{y}{x} = \tanθ \implies y = x\tanθ). So (\boxed{y = x \tanθ}) (if that's the target).

But given the input, the best answer is an explanation of the identities. However, if the user expects a boxed result (common in math problems), perhaps the key identity like (x^2 + y^2 = r^2) or (y = r\sinθ) is intended. Let's pick the most fundamental one:
(\boxed{x^2 + y^2 = r^2})
Or if the question was to confirm all are correct, then a statement like (\boxed{\text{All identities are valid}}) would work. But given the context, I think the explanation is more helpful, but since the user might want a boxed answer, I'll go with the core identity.

(\boxed{x^2 + y^2 = r^2})

作者声明:本文包含人工智能生成内容。