Suppose
L
is the tangent line of
y
=
3
tan
⁡
x
−
2
csc
⁡
x
at
x
=
π
3
.

Where does
L
intersect the
x
-axis?

Question

Suppose 
L
  is the tangent line of 
y
=
3
tan
⁡
x
−
2
csc
⁡
x
  at 
x
=
π
3
 .

Where does 
L
  intersect the  
x
-axis?

BlackTom AI · Accepted Answer

We begin by restating the problem in our own words to ensure we understand it: We have the function y = 3 tan x − 2 csc x, and L is the tangent line to this curve at x = π/3. We must find where this tangent line L intersects the x-axis (i.e., where y = 0). Now, step by step:

1) Compute the point of tangency (x0, y0) on the curve at x0 = π/3.
- tan(π/3) = √3, so 3 tan x at x0 is 3√3.
- csc(π/3) = 2/√3, so 2 csc x at x0 is 4/√3, and with the minus sign it becomes −4/√3.
- Therefore y0 = 3 tan(π/3) − 2 csc(π/3) = 3√3 − 4/√3 = (9 − 4)/√3 = 5/√3.
So the point of tangency is (x0, y0) = (π/3, 5/√3).

2) Find the slope m of the tangent line at x0.
- y = 3 tan x − 2 csc x.
- dy/dx = 3 sec^2 x − 2 d/dx[csc x]. Since d/dx[csc x] = −csc x cot x, we get dy/dx = 3 sec^2 x − 2(−csc x cot x) = 3 sec^2 x + 2 csc x cot x.
- Evaluate at x0 = π/3: cos(π/3) = 1/2, so sec^2(π/3) = 1/cos^2(π/3) = 1/(1/4) = 4, giving 3 sec^2 x term as 12.
- sin(π/3) = √3/2, so csc(π/3) = 2/√3 and cot(π/3) = cos(π/3)/sin(π/3) = (1/2)/(√3/2) = 1/√3. Thus 2 csc x cot x = 2*(2/√3)*(1/√3) = 4/3.
- Therefore m = 12 + 4/3 = 40/3.

3) Write the equation of the tangent line L at (x0, y0): y − y0 = m (x − x0).
So y − 5/√3 = (40/3) (x − π/3).

4) Find the x-intercept of L by setting y = 0 and solving for x.
- 0 − 5/√3 = (40/3) (x − π/3).
- −5/√3 = (40/3) (x − π/3).
- Multiply both sides by 3/40: x − π/3 = −(5/√3) · (3/40) = −15/(40√3) = −3/(8√3).
- Thus x = π/3 − 3/(8√3).
- If we rationalize 3/(8√3), we get 3/(8√3) · (√3/√3) = (3√3)/ (8·3) = √3/8. Hence x = π/3 − √3/8.

5) Compare with the provided answer choices.
- The listed options propose various expressions for the x-intercept, but none match the computed value x = π/3 − √3/8 exactly, since √3/8 ≈ 0.2165 while the form π/3 − 3/8 would use a numeric subtraction without the √3 factor, giving a different numerical result.

6) If we evaluate the provided options numerically for comparison:
- π/3 ≈ 1.0472. π/3 − 3/8 ≈ 1.0472 − 0.375 = 0.6722 (not equal to ~0.8307 that would result from using √3/8).
- Other options mix in unusual constants like 40π/3 or 40/3 fractions that do not align with the exact intercept we computed.

7) Conclusion about the intercept based on correct calculus (without concluding which listed option is correct): the precise x-coordinate of the intersection is x = π/3 − √3/8, which is equivalent to x = π/3 − (√3)/8 after simplification. The options given do not match this exact expression, so none of the listed choices precisely corresponde to the computed intercept. The mismatch highlights the importance of careful algebra when rationalizing expressions and keeping track of trigonometric values.

Note: If you need, I can show each algebraic step to convert between the form with √3 in the denominator and the rationalized form, or recalculate with an alternative method to verify the result.

MAT135H5_F25_ALL SECTIONS 3.5 Preparation Check

Suppose L is the tangent line of y = 3 tan ⁡ x − 2 csc ⁡ x at x = π 3 . Where does L intersect the x -axis?

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