FIND THE EQUATION OF THE TANGENT LINE OF TE FUNCTION AT THE GIVEN VALUE(S). A \( f(x)=4 \sin (x)-2 \) at \( x=\hat{\pi} \)

To find the equation of the tangent line to the function \( f(x) = 4 \sin(x) - 2 \) at \( x = \hat{\pi} \), we first need to compute the derivative \( f'(x) \). The derivative \( f'(x) = 4 \cos(x) \). Evaluating the derivative at \( x = \hat{\pi} \): \[ f'(\hat{\pi}) = 4 \cos(\hat{\pi}) = 4(-1) = -4. \] Next, we calculate \( f(\hat{\pi}) \): \[ f(\hat{\pi}) = 4 \sin(\hat{\pi}) - 2 = 4(0) - 2 = -2. \] Now we have the slope of the tangent line as -4 and the point at which the line touches the function as \((\hat{\pi}, -2)\). Using the point-slope form \( y - y_1 = m(x - x_1) \), we can write the equation of the tangent line: \[ y - (-2) = -4(x - \hat{\pi}) \] \[ y + 2 = -4x + 4\hat{\pi} \] \[ y = -4x + 4\hat{\pi} - 2. \] And there you have it! The equation of the tangent line at \( x = \hat{\pi} \) is given by \( y = -4x + 4\hat{\pi} - 2 \). To really cozy up to the concept of tangents, did you know that the concept dates back to ancient Greece? Mathematicians like Archimedes were fascinated by the properties of curves and how they relate to lines. They laid the groundwork for modern calculus, so every time you find a tangent line, give a nod to those early thinkers! Fun fact: Tangent lines aren't just for curves! They play a crucial role in areas like physics, specifically in finding instantaneous velocity. When you're driving and you want to know your speed at a precise moment, you're essentially looking for the slope of the curve representing your position over time – the tangent line to your position function! Isn't that a cool way to see the world?