If `ycost" and "x=sint,` then what is the value of `(dy)/(dx)` ?

Options

A. xy
B. x/y
C. `-y//x`
D. `-x//y`

Answer

To find the value of \(\frac{dy}{dx}\) given that \(y = \cos t\) and \(x = \sin t\), we can follow these steps:

### Step 1: Differentiate \(y\) with respect to \(t\)
We start by differentiating \(y\) with respect to \(t\):
\[ \frac{dy}{dt} = \frac{d}{dt}(\cos t) = -\sin t \]

### Step 2: Differentiate \(x\) with respect to \(t\)
Next, we differentiate \(x\) with respect to \(t\):
\[ \frac{dx}{dt} = \frac{d}{dt}(\sin t) = \cos t \]

### Step 3: Use the chain rule to find \(\frac{dy}{dx}\)
Using the chain rule, we can express \(\frac{dy}{dx}\) in terms of \(\frac{dy}{dt}\) and \(\frac{dx}{dt}\):
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \]

### Step 4: Substitute the derivatives from Steps 1 and 2
Now we substitute the values we found in Steps 1 and 2:
\[ \frac{dy}{dx} = \frac{-\sin t}{\cos t} \]

### Step 5: Simplify the expression
We can simplify \(\frac{-\sin t}{\cos t}\) to:
\[ \frac{dy}{dx} = -\tan t \]

### Step 6: Relate \(\sin t\) and \(\cos t\) to \(x\) and \(y\)
Since \(x = \sin t\) and \(y = \cos t\), we can express \(\frac{dy}{dx}\) in terms of \(x\) and \(y\):
\[ \frac{dy}{dx} = -\frac{x}{y} \]

### Final Result
Thus, the value of \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = -\frac{x}{y} \]