find the points on the graph of y=secx where the tangent is parrallel to 6y-4x=8. (0 is less than or equal to x and x is less than or equal to 2pi). Use exact values.
To find the points on the graph of y = secx where the tangent is parallel to the line 6y - 4x = 8, we follow these steps:
Step 1: Find the slope of the given line.
To find the slope, we can rewrite the given equation in slope-intercept form:
6y - 4x = 8
6y = 4x + 8
y = (4/6)x + 8/6
y = (2/3)x + 4/3
Comparing this equation with y = mx + b, we can see that the slope (m) of the line is 2/3.
Step 2: Find the derivative of y = secx.
The derivative of y = secx can be found by applying the chain rule.
dy/dx = secx * tanx
This derivative represents the slope of the tangent line at any given point on the graph of y = secx.
Step 3: Set the derivative equal to the slope of the given line.
Since we want the tangent line to be parallel to the line with slope 2/3, we set the derivative of y = secx equal to 2/3, as follows:
secx * tanx = 2/3
Step 4: Solve the equation for x.
To solve the equation secx * tanx = 2/3, we can use trigonometric identities.
tanx = sinx / cosx
secx = 1 / cosx
Substituting these identities into the equation, we get:
(1/cosx) * (sinx / cosx) = 2/3
sinx / cos^2x = 2/3
sinx = (2/3) * cos^2x
Now, we can use the Pythagorean identity sin^2x + cos^2x = 1 to solve for cosx.
sin^2x + cos^2x = 1
(2/3)^2 * cos^4x + cos^2x = 1
(4/9) * cos^4x + cos^2x = 1
Let's make a substitution, where u = cos^2x.
(4/9) * u^2 + u = 1
4u^2 + 9u - 9 = 0
Solving this quadratic equation for u, we get:
(4u - 3)(u + 3) = 0
Solving for u, we have two possible solutions:
4u - 3 = 0 --> u = 3/4
u + 3 = 0 --> u = -3 (discarded since cos^2x cannot be negative)
Therefore, the possible values for cosx are:
cosx = sqrt(3)/2 and cosx = -sqrt(3)/2
Step 5: Solve for x using the inverse trigonometric function.
The values of cosx allow us to find the corresponding values of x using the inverse trigonometric function.
For cosx = sqrt(3)/2:
x = arccos(sqrt(3)/2)
x = pi/6 (since arccos(sqrt(3)/2) = pi/6)
For cosx = -sqrt(3)/2:
x = arccos(-sqrt(3)/2)
x = 5pi/6 (since arccos(-sqrt(3)/2) = 5pi/6)
Step 6: Determine the corresponding values of y.
Using the values of x obtained in Step 5, we can find the corresponding values of y by substituting them into the equation y = secx.
For x = pi/6:
y = sec(pi/6)
y = 2
For x = 5pi/6:
y = sec(5pi/6)
y = 2
Therefore, the points on the graph of y = secx where the tangent is parallel to 6y - 4x = 8 are (pi/6, 2) and (5pi/6, 2).