What angle does the tangent to the curve y(x)=sin^2(x/3)+15sin(x/3) at x=3pi make with the x-axis?
dy/dx = 2(1/3)sin(x/3) cos(x/3) + (1/3)(15)cos(x/3)
at x = 3π
dy/dx = (2/3)sinπ cosπ + 5cosπ
= 0 + 5(-1) = -5
angle that tangent makes is tan^-1 (-5) = 101.3°
I'd rather pick -78.7°
As x is increasing, y is decreasing.
Also, the principal values of tan^-1 are -π/2 to π/2
To find the angle that the tangent to the curve makes with the x-axis, we need to find the derivative of the curve and evaluate it at the given point.
The given curve is y(x) = sin^2(x/3) + 15sin(x/3).
Step 1: Find the derivative of y(x) with respect to x.
Let's use the chain rule to find the derivative of the first term, sin^2(x/3), and the second term, 15sin(x/3).
For the first term, we have:
d/dx [sin^2(x/3)] = 2sin(x/3) * (d/dx)[sin(x/3)]
= 2sin(x/3) * (1/3)cos(x/3) (applying chain rule)
= (2/3) sin(x/3) cos(x/3)
For the second term, we have:
d/dx [15sin(x/3)] = 15 * (d/dx)[sin(x/3)]
= 15 * (1/3)cos(x/3) (applying chain rule)
= 5 cos(x/3)
Therefore, the derivative of y(x) is:
dy/dx = (2/3) sin(x/3) cos(x/3) + 5 cos(x/3)
= (2/3) sin(x/3) cos(x/3) + (15/3) cos(x/3)
= (2/3) sin(x/3) cos(x/3) + (5/1) cos(x/3)
= (2/3 + 5/1) cos(x/3) + (2/3) sin(x/3) cos(x/3)
= (17/3) cos(x/3) + (2/3) sin(x/3) cos(x/3)
= (17/3) cos(x/3) + (2/3) sin^2(x/3)
Step 2: Evaluate the derivative at x = 3π.
To find the slope of the tangent line at x = 3π, we need to evaluate the derivative at this point.
dy/dx = (17/3) cos(x/3) + (2/3) sin^2(x/3)
Substituting x = 3π, we have:
dy/dx = (17/3) cos((3π)/3) + (2/3) sin^2((3π)/3)
= (17/3) cos(π) + (2/3) sin^2(π)
= (17/3) (-1) + (2/3) (0)
= -17/3
Step 3: Find the tangent line's slope in relation to the x-axis.
The tangent line's slope in relation to the x-axis is the negative reciprocal of the derivative.
Let's call the slope of the tangent line in relation to the x-axis as m'. Then:
m' = -1/(dy/dx)
= -1/(-17/3)
= 3/17
Step 4: Find the angle that the tangent line makes with the x-axis.
The angle that the tangent line makes with the x-axis, θ, can be found using the inverse tangent function.
θ = arctan(m')
= arctan(3/17)
Therefore, the angle that the tangent to the curve makes with the x-axis is approximately 9.44 degrees.
To find the angle that the tangent to the curve makes with the x-axis, you need to determine the slope of the tangent line at the given point x = 3π.
First, find the first derivative of the given function y(x) = sin²(x/3) + 15sin(x/3) to find the slope of the tangent line.
Differentiating y(x) with respect to x:
dy/dx = [2sin(x/3) * (1/3)] + [15cos(x/3) * (1/3)]
Next, substitute x = 3π into the derived equation, as you want to find the slope at x = 3π:
dy/dx at x = 3π = [2sin(3π/3) * (1/3)] + [15cos(3π/3) * (1/3)]
Simplifying further:
dy/dx at x = 3π = (2 * sin(π)) / 3 + (15 * cos(π)) / 3
Since sin(π) = 0 and cos(π) = -1, the equation becomes:
dy/dx at x = 3π = (2 * 0) / 3 + (15 * -1) / 3
dy/dx at x = 3π = -15/3
dy/dx at x = 3π = -5
Now, the slope of the tangent line at x = 3π is -5. To find the angle that the tangent line makes with the x-axis, you can use the trigonometric identity:
tan(θ) = m
where θ is the angle and m is the slope. Rearranging the equation:
θ = arctan(m)
Substituting the value of the slope (-5):
θ = arctan(-5)
Using a calculator or math software, you can find the value of this angle to be approximately -78.69 degrees or -1.37 radians.
Therefore, the angle that the tangent to the curve y(x) = sin²(x/3) + 15sin(x/3) at x = 3π makes with the x-axis is approximately -78.69 degrees or -1.37 radians.