How do you find at equation of the tangent line to the graph #y=(cosx)/(sinx+2)# at x =pi/2?

Question

Samantha Adkison · Accepted Answer

#y = pi/6-1/3x#

Start by finding the y-coordinate of tangency.

#y = cos(pi/2)/(sin(pi/2) + 2)#

#y= 0/(1 + 2)#

#y = 0#

Now differentiate using the quotient rule.

#y' = (-sinx(sinx + 2) - cosx(sinx))/(2 + sinx)^2#

#y' = (-sin^2x - 2sinx - cosxsinx)/(2 + sinx)^2#

The slope of the tangent is given by evaluating your point #x= a# into the derivative.

#m_"tangent" = (-sin^2(pi/2) - 2sin(pi/2) - cos(pi/2)sin(pi/2))/(2+ sin(pi/2))^2#

#m_"tangent" = (-(1)^2 - 2(1) - 0(1))/(2 + 1)^2#

#m_"tangent" = (-1 - 2)/(2 + 1)^2#

#m_"tangent"= -3/9=-1/3#

We now find the equation of the tangent.

#y - y_1 = m(x - x_1)#

#y - 0 = -1/3(x - pi/2)#

#y = pi/6-1/3x #

Hopefully, this helps!

William Abdalla · Answer

undefined To find the equation of the tangent line to the graph of y = (cosx)/(sinx+2) at x = pi/2, we can follow these steps:

1. Find the derivative of the function y = (cosx)/(sinx+2) using the quotient rule.
2. Evaluate the derivative at x = pi/2 to find the slope of the tangent line.
3. Use the point-slope form of a line, y - y1 = m(x - x1), where (x1, y1) is the point of tangency and m is the slope, to find the equation of the tangent line.

Let's go through these steps:

1. Differentiating y = (cosx)/(sinx+2) using the quotient rule:
   - The derivative of the numerator, cosx, is -sinx.
   - The derivative of the denominator, sinx+2, is cosx.
   - Applying the quotient rule: (cosx * (sinx+2) - (cosx * cosx)) / (sinx+2)^2.
   - Simplifying the numerator: (cosx * sinx + 2cosx - cos^2x) / (sinx+2)^2.

2. Evaluating the derivative at x = pi/2:
   - Substitute x = pi/2 into the derivative expression obtained in step 1.
   - Simplifying the expression: (-1 * 1 + 2 * 0 - cos^2(pi/2)) / (sin(pi/2)+2)^2.
   - Since cos(pi/2) = 0 and sin(pi/2) = 1, the expression becomes: (-1 - 0) / (1+2)^2 = -1/9.

3. Using the point-slope form of a line:
   - The point of tangency is (pi/2, (cos(pi/2))/(sin(pi/2)+2)).
   - Substituting the values into the point-slope form: y - ((cos(pi/2))/(sin(pi/2)+2)) = (-1/9)(x - pi/2).
   - Simplifying the equation: y - (1/2) = (-1/9)(x - pi/2).

Therefore, the equation of the tangent line to the graph y = (cosx)/(sinx+2) at x = pi/2 is y - (1/2) = (-1/9)(x - pi/2).