Question

Convert the following equations to Cartesian coordinates. Describe the resulting curve. $$\sin \theta=|\cos \theta|$$

Short answer

Question: Convert the polar equation $\sin \theta = |\cos \theta|$ to Cartesian coordinates and describe the resulting curve. Answer: The corresponding Cartesian equation is $y^2 = 0$. The resulting curve is a horizontal line at y = 0, which is the x-axis itself.

Step-by-step solution

Recall polar and Cartesian coordinates relationship

The relationship between polar coordinates (r, θ) and Cartesian coordinates (x, y) is given by: $$x = r\cos\theta$$ $$y = r\sin\theta$$ We will use these relationships to convert the given polar equation to Cartesian coordinates. Step 2: Solving for r in terms of θ

Solve for r in terms of θ

We are given the polar equation: $$\sin \theta = |\cos \theta|$$ Square both sides of the equation to eliminate the absolute value: $$(\sin^2\theta) = (\cos^2\theta)$$ Add sin^2θ to both sides, so that we have a Pythagorean identity relation: $$r^2(\sin^2\theta + \cos^2\theta) = r^2\cos^2\theta$$ Step 3: Substitute the polar-Cartesian relationship

Substitute the polar-Cartesian relationship

Now, we can use the polar-Cartesian relationship x = rcosθ and y = rsinθ to rewrite the equation in Cartesian coordinates. We have: $$r^2 = r^2\cos^2\theta$$ Substitute x and y into this equation: $$(x^2+y^2) = x^2$$ Step 4: Simplify the equation

Simplify the Cartesian equation

Now, simplify the equation by subtracting x^2 from both sides to obtain the equation in Cartesian coordinates: $$y^2=x^2-x^2$$ $$y^2=0$$ Step 5: Describe the resulting curve

Describe the curve

The resulting Cartesian equation is y^2 = 0, which represents a curve along the x-axis. In other words, the curve is a horizontal line at y = 0, which is the x-axis itself.