Answer:
Tangent: [tex]y = (-2)\, x + (2\, \pi - 1)[/tex].
Normal: [tex]y = (1/2)\, x + ((-1/2)\, \pi - 1)[/tex].
Step-by-step explanation:
The tangent to the curve at the given point can be found in the following steps:
The curve in this question is given as a function of [tex]x[/tex]: [tex]f(x) = \cos(x) + 2\, \sin(x)[/tex]. To find the slope of this curve at [tex]x = \pi[/tex], find an expression for the first derivative of [tex]f(x)[/tex] and evaluate that expression at [tex]x = \pi[/tex].
[tex]\begin{aligned} f^{\prime}(x) &= \frac{d}{d x}\left[\cos(x) + 2\, \sin(x)\right] \\ &= -\sin(x) + 2\, \cos(x)\end{aligned}[/tex].
Substitute in [tex]x = \pi[/tex] and evaluate to obtain:
[tex]f^{\prime}(\pi) = -\sin(\pi) + 2\, \cos(\pi) = 0 + 2\, (-1) = -2[/tex].
In other words, the slope of the curve at the given point is [tex](-2)[/tex].
Evaluate the function describing this curve [tex]f(x) = \cos(x) + 2\, \sin(x)[/tex] at [tex]x = \pi[/tex] to find the coordinates of the requested point:
[tex]f(\pi) = \cos(\pi) + 2\, \sin(\pi) = (-1) + 0 = (-1)[/tex].
In other words, the specified point on the curve would be [tex](\pi,\, -1)[/tex]. The tangent line should go through this point. Since it is deduced that the slope of the tangent line is [tex](-2)[/tex], the point-slope equation of this tangent line would be:
[tex]\displaystyle y - (-1) = (-2)\, (x - \pi)[/tex].
Simplify to obtain the slope-intercept equation of this line:
[tex]y = (-2)\, x + (2\, \pi - 1)[/tex].
The normal to the curve at the given point is perpendicular to the tangent. If two lines on a cartesian plane are perpendicular to one another, the product of their slopes should be [tex](-1)[/tex]. Since the slope of the tangent line at this point is [tex](-2)[/tex], the slope of the normal would be: [tex](-1) / (-2) = (1/2)[/tex].
The normal would also go through the point [tex](\pi,\, -1)[/tex]. Hence, the point-slope equation of the normal would be:
[tex]\displaystyle y - (-1) = \frac{1}{2}\, (x - \pi)[/tex].
Simplify to obtain the slope-intercept equation of this line:
[tex]\displaystyle y = \frac{1}{2}\, x+ \left(-\frac{1}{2}\,\pi - 1\right)[/tex].
