Find the equation of the tangents to the curve \( y=\cos x \) at \( x=-\pi, \frac{\pi}{2} \), and \( -\frac{3 \pi}{2} \). Graph the curve over the interval \( \left[-\frac{3 \pi}{2}, 2 \pi\right] \) together with their tangents. Label each curve and tangent. What is the equation of the tangent \( (1) \) to the curve at \( x=-\pi \) ? \( y=\square \)

To find the equation of the tangent to the curve \( y = \cos x \) at a given point, we need to follow these steps: 1. **Find the derivative of the function** to determine the slope of the tangent line. 2. **Evaluate the function and the derivative at the specified point** to find the coordinates of the point of tangency and the slope. 3. **Use the point-slope form of the equation of a line** to write the equation of the tangent line. ### Step 1: Find the derivative The derivative of \( y = \cos x \) is: \[ \frac{dy}{dx} = -\sin x \] ### Step 2: Evaluate at \( x = -\pi \) 1. **Find the function value**: \[ y(-\pi) = \cos(-\pi) = -1 \] 2. **Find the slope**: \[ \frac{dy}{dx} \bigg|_{x=-\pi} = -\sin(-\pi) = 0 \] So, at the point \( (-\pi, -1) \), the slope of the tangent line is \( 0 \). ### Step 3: Write the equation of the tangent line Using the point-slope form \( y - y_1 = m(x - x_1) \): - Point: \( (-\pi, -1) \) - Slope: \( 0 \) The equation becomes: \[ y - (-1) = 0(x - (-\pi)) \] \[ y + 1 = 0 \] \[ y = -1 \] Thus, the equation of the tangent at \( x = -\pi \) is: \[ y = -1 \] ### Summary The equation of the tangent (1) to the curve at \( x = -\pi \) is: \[ y = -1 \] ### Next Steps To find the equations of the tangents at \( x = \frac{\pi}{2} \) and \( x = -\frac{3\pi}{2} \), we can follow the same steps: #### For \( x = \frac{\pi}{2} \): 1. **Function value**: \[ y\left(\frac{\pi}{2}\right) = \cos\left(\frac{\pi}{2}\right) = 0 \] 2. **Slope**: \[ \frac{dy}{dx} \bigg|_{x=\frac{\pi}{2}} = -\sin\left(\frac{\pi}{2}\right) = -1 \] 3. **Equation**: \[ y - 0 = -1\left(x - \frac{\pi}{2}\right) \] \[ y = -x + \frac{\pi}{2} \] #### For \( x = -\frac{3\pi}{2} \): 1. **Function value**: \[ y\left(-\frac{3\pi}{2}\right) = \cos\left(-\frac{3\pi}{2}\right) = 0 \] 2. **Slope**: \[ \frac{dy}{dx} \bigg|_{x=-\frac{3\pi}{2}} = -\sin\left(-\frac{3\pi}{2}\right) = 1 \] 3. **Equation**: \[ y - 0 = 1\left(x + \frac{3\pi}{2}\right) \] \[ y = x + \frac{3\pi}{2} \] ### Final Tangent Equations 1. At \( x = -\pi \): \( y = -1 \) 2. At \( x = \frac{\pi}{2} \): \( y = -x + \frac{\pi}{2} \) 3. At \( x = -\frac{3\pi}{2} \): \( y = x + \frac{3\pi}{2} \) ### Graphing To graph the curve \( y = \cos x \) and the tangents over the interval \( \left[-\frac{3\pi}{2}, 2\pi\right] \), you can use graphing software or tools to visualize the function and the tangent lines. Each curve and tangent should be labeled accordingly.